Les vecteurs cycliques dans des espaces de fonctions

Les vecteurs cycliques dans des espaces de fonctions

analytiques

Abdelouahab Hanine

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Abdelouahab Hanine. Les vecteurs cycliques dans des espaces de fonctions analytiques. Analysefonctionnelle [math.FA]. Aix-Marseille Universite, 2013. Francais. <tel-00965831>

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Spécialité : Mathématiques

École doctorale de Mathématiques et informatiques de Marseille

présentée par

Abdelouahab hanine

pour l’obtention du

DIPLOME DE DOCTORAT

Sujet :

Cyclic vectors in some spaces of analyticfunctions

Directeurs de thèse :

M. Alexander BORICHEV, Professeur à l’Aix Marseille université.M. Omar EL-FALLAH, P.E.S à la faculté des sciences de Rabat.

Thèse soutenue publiquement le 28 juin 2013 devant le jury composé de :

M. A. Baalal Univ. Hassan II, Casablanca, Maroc RapporteurM. A. Borichev Aix Marseille université DirecteurM. J. Conway George Washington University, USA ExaminateurM. O. El-Fallah Univ. Mohamed V, Rabat, Maroc DirecteurM. K. Kellay Université Bordeaux I ExaminateurM. A. Montès-Rodriguès Université de Seville, Espagne ExaminateurM. P. Thomas Université Paul Sabatier RapporteurM. E. H. Youssfi Aix Marseille université ExaminateurM. E. H. Zerouali Univ. Mohamed V, Rabat, Maroc Examinateur

Cyclic vectors in some spaces ofanalytic functions

A DissertationPresented for the Doctor of Philosophy Degree

The University of Aix-Marseille andthe University of Mohammed V-Agdal, Rabat

Abdelouahab Hanine

18 juin 2013

Remerciements

Cette thèse a été élaborée en cotutelle entre le Laboratoire d’Analyse Topologie et Pro-babilités d’Aix-Marseille Université et le Laboratoire d’Analyse et Applications de l’Uni-versité Mohamed V- Rabat-Agdal.

Mes premiers remerciements vont à mes directeurs de thèse, le professeur AlexanderBorichev et le professeur Omar El-Fallah, sans qui cette thèse n’aurait pas pu exister,progresser et aboutir. Je suis très heureux d’avoir pu profiter pendant ces années de sesgrandes cultures mathématiques. Je les remercie de m’avoir toujours laissé cette part d’au-tonomie sur les problèmes que je voulais traiter et pour m’avoir fouetté afin que je travailleplus vite et plus efficacement ! Cette dernière n’aurait pas vu le jour sans la confiance, lapatience, et la générosité qu’ils ont su m’accorder. Ce fut un réel plaisir de partager toutesces heures à vos côtés sur des problèmes passionnants. J’espère que nous continuerons àtravailler ensemble.

Je tiens à exprimer ma reconnaissance à Azeddine Baalal et Pascal Thomas d’avoiraccepté de juger ce travail, pour les remarques qu’ils m’ont faites et pour leur efficacité.

Je tiens à exprimer ma gratitude aux Professeurs, John Conway, Karim Kellay, AlfonsoMontès-Rodriguès, El Hassan Youssfi, et El Hassan Zerouali qui ont accepté de faire partiede mon jury.

Je souhaiterais également exprimer toute ma reconnaissance à certains de mes pro-fesseurs qui, tant par leurs qualités d’enseignant que par leur gentillesse, ont marqué mascolarité et pour qui j’ai une affection particulière. Je pense en particulier à Azzedine Daouiqui m’a fait partagé avec beaucoup de générosité et de simplicité son goût des mathéma-tiques en dirigeant mon PFE et à Azeddine Baalal dont j’admire le travail, l’enthousiasmeet le dévouement pour ses élèves. Je les remercie tous les deux pour leurs encouragementsqui m’ont énormément aidé.

Je remercie chaleureusement toutes les personnes du CMI et du département de ma-thématiques de la faculté des sciences de Rabat qui m’ont toujours reçue avec beaucoupde gentillesse lorsque j’avais des problèmes d’ordre administatif, informatique.

Je remercie mes parents et mes frères et mes sœurs pour leur soutien pendant ceslongues années d’étude.

Je remercie également mes amis pour leur encouragements et le soutien très précieuxqu’ils m’ont apporté.

1

Abstract

In this thesis, we study the cyclicity problem in some spaces of analytic functions onthe open unit disc. We focus our attention on Korenblum type spaces and on weightedBergman type spaces.

First, we use the technique of premeasures, introduced and developed by Korenblum inthe 1970-s and the 1980-s, to give a characterization of cyclic functions in the Korenblumtype spaces A−∞

Λ . In particular, we give a positive answer to a conjecture by Deninger.Second, we use the so called resolvent transform method to study the cyclicity of theone point mass singular inner function S(z) = e−

1+z1−z in weighted Bergman type spaces,

especially with weights depending on the distance to a subset of the unit circle.

2

Résumé de la thèse

Cette thèse est consacrée à l’étude du problème de la cyclicité dans certains espaces defonctions analytiques. Nous nous intéressons aux espaces de Bergman et aux espaces detype Korenblum. La caractérisation des fonctions cycliques dans les espaces de fonctionsanalytiques est un problème d’approximation polynomiale.

On désignera par D le disque unité ouvert dans le plan complexe C et par T le cercleunité de C. L’ensemble des fonctions holomorphes sur D sera noté Hol(D).

Soit X un espace vectoriel topologique de fonctions analytiques sur D. Soit Mz l’opé-rateur de multiplication par la variable z défini par

Mz(f) = zf, f ∈ X.

On suppose dans la suite de cette section, que Mz définit un opérateur borné sur X. On ditqu’un sous-espace fermé M de X (espace de Banach) est invariant par Mz (ou z-invariant)si zM ⊂ M. Pour une fonction f ∈ X, nous notons [f ]X le plus petit sous-espace ferméde X contenant f et invariant par Mz. Il est alors clair que

[f ]X :=p(z)f(z) : p(z) polynôme

X,

(on considère la fermeture dans X).Une fonction f ∈ X est dite cyclique dans X lorsque [f ]X = X.Supposons que les fonctionnelles d’évaluations sur X définies par :

f 7→ f(z), z ∈ D,

sont bornées. Il est alors clair que pour qu’une fonction f soit cyclique dans X, il faut quef n’ait pas de zéros dans D. Cette condition n’est pas suffisante en général. En effet, siX = H2 l’espace de Hardy, alors d’après le théorème de Beurling [4], f ∈ H2 est cycliquesi et seulement si f est extérieure, c’est-à-dire ils existent h ∈ L1(T) et |λ| = 1 tels que

f(z) = λ exp( 1

2π

∫

T

eit + z

eit − zh(eit) dt

).

Espaces de type Korenblum

Définitions et généralités

Dans la suite, le majorant Λ désigne toujours une fonction positive, différentiable, stric-tement décroissante et convexe sur (0,1] telle que :

3

1. Λ(0) = +∞

2. tΛ(t) est une fonction continue, strictement croissante et concave sur [0, 1] et tΛ(t) →0 quand t→ 0.

3. Il existe α ∈ (0, 1) tel que la fonction tαΛ(t) est strictement croissante.

4. il existe C > 0, tel que pour tout t ∈ (0, 1) on a

Λ(t2) ≤ CΛ(t). (0.1)

Nous pouvons prendre comme exemples : Λ(t) = log+ log+(1/t), Λ(t) = (log(1/t))p, p > 0.L’espace de type Korenblum associé au poids Λ est donné par

A−∞Λ = ∪c>0A

−cΛ =

⋃

c>0

f ∈ Hol(D) : |f(z)| ≤ exp(cΛ(1− |z|))

.

Dans ce travail nous étudions la cyclicité des éléments de A−∞Λ . Muni de la topologie

limite inductive et de la multiplication ponctuelle l’espace A−∞Λ devient une algèbre de

Fréchet. Notons que les polynômes sont denses dans A−∞Λ , et que Mz et les évaluations

sont continues sur A−∞Λ .

Prémesures Λ-bornées

B. Korenblum a étudié les prémesures Λ-bornées pour Λ(t) = log(1t). Dans cette section,

nous étendons les résultats des deux papiers de B. Korenblum [31, 32] (voir aussi [24,Chapitre 7]) aux prémesures Λ-bornées, pour une fonction Λ vérifiant les conditions ci-dessus.

On note B(T) l’ensemble de tous les arcs ouverts, fermés et semi-ouverts de T. Parconvention on suppose aussi que ∅,T ∈ B(T).

Définition 1 Une prémesure µ est une fonction réelle définie sur B(T) est vérifiant

1. µ (T) = 0

2. µ (I1 ∪ I2) = µ (I1)+µ (I2) pour tout I1, I2 ∈ B (T) avec I1∩ I2 = ∅ et I1∪I2 ∈ B (T)

3. limn→+∞

µ(In) = 0, pour toute suite décroissante (In)n ∈ B (T) avec⋂n In = ∅

Dans la suite, nous noterons |I| la mesure de Lebesgue normalisée de I.

Définition 2 Une prémesure µ est dite Λ-bornée s’il existe une constante positive C ≥ 0telle que pour tout arc I ⊂ T,

µ(I) ≤ C|I|Λ(|I|).

On note B+Λ l’ensemble des prémesures Λ-bornées, et par ‖µ‖+ la plus petite constante

C > 0 vérifiant la condition :µ(I) ≤ C|I|Λ(|I|).

4

Soit F un sous-ensemble fermé non vide du cercle unité T. L’entropie de F associée àΛ (Λ-entropie) est définie par :

EntrΛ(F ) =∑

n

|In|Λ(|In|),

où Inn sont les composantes connexes de T \ F, et |I| désigne la mesure de Lebesguenormalisée de l’arc I. On pose EntrΛ(∅) = 0.

Un ensemble F fermé non vide est dit ensemble Λ-Carleson s’il est de mesure de Le-besgue nulle (|F | = 0) et EntrΛ(F ) < +∞.

On note par CΛ l’ensemble de tous les ensembles Λ-Carleson et par BΛ l’ensemble detous les boréliens de T (B ⊂ T) tels que B ∈ CΛ.

Nous pouvons maintenant introduire la notion de mesure Λ-singulière.

Définition 3 Une fonction σ : BΛ → R est dite mesure Λ-singulière si

1. σ∣∣F

s’étend en une mesure de Borel finie sur T, pour tout F ∈ CΛ, (σ∣∣F(E) =

σ(E ∩ F )) pour tout fermé de T.

2. Il existe C > 0 tel que|σ(F )| ≤ CEntrΛ(F )

pour tout F ∈ CΛ.

Étant donnée µ une prémesure Λ-bornée, on dénote

µs(F ) = −∑

n

µ(In), (0.2)

où F ∈ CΛ et les Inn sont les arcs complémentaires de F dans le cercle T. En utilisantle même argument que dans [31, Théorème 6], on peut voir que µs s’étend en une mesureΛ-singulière sur BΛ. La mesure µs est appelée la partie Λ-singulière de µ.

Nous allons introduire maintenant la notion de prémesure Λ-absolument continue, quijouera un rôle important dans la caractérisation des fonctions cycliques de A−∞

Λ .

Définition 4 Une prémesure Λ-bornée est dite Λ-absolument continue s’il existe une suitede prémesures Λ-bornées (µn)n telle que :

1. (µ+ µn) ∈ B+Λ et sup

n‖µ+ µn‖

+ <∞.

2. supI |(µ+ µn)(I)| → 0 quand n→ +∞.

Dans le cas des mesures, µ est absolument continue si et seulement si µ n’a pas departie singulière. Ce résultat s’étend au cas des prémesures Λ-bornées, nous obtenons lerésultat suivant.

Théorème 1 Une prémesure µ ∈ B+Λ est Λ-absolument continue si et seulement si µs ≡ 0.

La preuve est détaillée dans [21, Section 2.8]

5

Fonctions harmoniques à croissance contrôlé par Λ

Rappelons qu’une fonction harmonique bornée sur D peut être représentée par l’inté-grale de Poisson d’une fonction définie sur T. Dans le théorème suivant nous montronsqu’une large classe de fonctions harmoniques à valeurs réelles dans le disque unité D peutêtre représentée par les intégrales de Poisson de prémesures Λ-bornées.

Soit µ ∈ B+Λ , nous notons P [µ] le noyau de Poisson de µ,

P [µ](z) =

∫ 2π

0

1− |z|2

|eiθ − z|2dµ(θ),

et nous définissons l’intégrale en terme de prémesures Λ-bornées par :

P [µ](z) := −

∫ 2π

0

( ∂

∂θ

1− |z|2

|eiθ − z|2

)fµ(θ) dθ,

telle quefµ(θ) = µ(Iθ),

oùIθ =

ξ ∈ T : 0 ≤ arg ξ < θ

.

Théorème 2 Soit h une fonction harmonique sur D, à valeurs dans R avec h(0) = 0 telleque

h(z) = O(Λ(1− |z|)), |z| → 1, z ∈ D.

Les assertions suivantes sont vérifiées :

1. Pour tout arc I du cercle unité T, la limite suivante existe

µ(I) = limr→1−

µr(I) = limr→1−

∫

I

h(rξ) |dξ| <∞.

2. µ est une prémesure Λ-bornée.

3. La foction h admet la représentation suivante (l’intégrale de Poisson de prémesureµ) :

h(z) =

∫ 2π

0

1− |z|2

|eiθ − z|2dµ(θ), z ∈ D.

Réciproquement, nous montrons que le noyau de Poisson d’une prémesure Λ-bornée estune fonction harmonique à croissance contrôlée par Λ, plus précisément

P [µ](z) ≤ 10‖µ‖+ΛΛ(1− |z|), z ∈ D.

Grâce à ce résultat, on peut représenter les fonctions de A−∞Λ qui ne s’annulent pas dans

D comme les intégrales de Poisson de prémesures Λ-bornées. En effet, si f ∈ A−∞Λ , f(0) = 1,

il existe une prémesure Λ-bornée µf (I) := µ(I) = limr→1−


∫Ilog |f(rξ)| |dξ|, I ∈

B(T) (voir le théorème 2) telle que :

f(z) = fµ(z) := exp

∫ 2π

0

eiθ + z

eiθ − zdµ(θ). (0.3)

6

Cyclicité

Le résultat suivant découle du théorème 1.

Théorème 3 Soit f ∈ A−∞Λ une fonction sans zéros sur D telle que f(0) = 1. Si (µf )s ≡ 0,

alors f est cyclique dans A−∞Λ .

Dans ce qui suit, nous traitons la réciproque du théorème 3. Nous établissons un résultatprincipal valable pour deux type de croissance de la fonction Λ. Plus précisément, nousconsidérons les deux cas suivants :

Cas 1. Nous supposons que le majorant Λ vérifie la condition (C1), définie par :

pour toute c > 0, x 7→ exp[cΛ(1/x)

]est une fonction concave pour x grand. (C1)

Exemple du majorant Λ qui vérifie la condition (C1) :

(log(1/x))p, 0 < p < 1, and log(log(1/x)), x→ 0.

Cas 2. Nous supposons que le majorant Λ vérifie la condition (C2), définie par :

limt→0

Λ(t)

log(1/t)= ∞. (C2)

Exemple du majorant Λ qui vérifie la condition (C2) :

(log(1/x))p, p > 1.

Nous obtenons le résultat suivant :

Théorème 4 Soit µ ∈ B+Λ . Alors la fonction fµ est cyclique dans A−∞

Λ si et seulement siµs ≡ 0.

Pour la démonstration de ce théorème, nous distinguons trois cas :Si Λ vérifie la condition (C1), nous utilisons théorème de Shirokov (voir [46, Théorème 9,

pp. 137,139]).Si Λ(t) = log(1

t) c’est le théorème de B. Korenblum [32, Théorème 3.1].

Si Λ vérifie la condition (C2), nous utilisons le théorème de Taylor et Williams (voir[11, Théorème 5.3]).

Remarquons que dans le cas 1, nous considérons des poids à croissance lente par rapportau poids de Korenblum (Λ(x) = log(1/x)) et dans le cas 2, la croissance de Λ est plus rapideque celle de Korenblum.

Le théorème 4, donne une réponse positive à une conjecture énoncée par C. Deninger[14, Conjecture 42].

Nous donnons maintenant deux exemples qui montrent comment la cyclicité d’unefonction fixée change en changeant le majorant Λ dans l’espace A−∞

Λ .

7

Exemple 1 Soit Λα(x) = (log(1/x))α, 0 < α < 1, et soit 0 < α0 < 1. Il existe unefonction singulière intérieure Sµ telle que

Sµ est cyclique dans A−∞Λα

⇐⇒ α > α0.

Exemple 2 Soit Λα(x) = (log(1/x))α, 0 < α < 1, et soit 0 < α0 < 1. Il existe uneprémesure µ telle que µs est infinie et

fµ est cyclique dans A−∞Λα

⇐⇒ α > α0,

où fµ est définie par la formule dans (0.3).

Il faut remarquer que le sous-espace [fµ]A−∞Λα

, α ≤ α0, est un sous-espace fermé non

trivial de A−∞Λα

. De plus, il ne contient aucune fonction non nulle de la classe de NevanlinnaN := f ∈ Hol(D) : sup0≤r<1

∫Tlog+ |f(reiθ)| dθ

2π<∞.

Cyclicité dans les espaces de type Bergman

Dans cette partie, nous étudions la cyclicité de la fonction intérieure singulière S(z) =e−

1+z1−z dans les espaces de type Bergman.Étant donnée une fonction positive continue strictement décroissante Λ sur (0, 1] et

E ⊂ T = ∂D, on note B∞Λ,E l’espace des fonctions analytiques f sur D telles que

‖f‖Λ,E,∞ = supz∈D

|f(z)|e−Λ(dist(z,E)) < +∞,

et par B∞,0Λ,E son sous-espace séparable

B∞,0Λ,E =

f ∈ B∞

Λ,E : limdist(z,E)→0

|f(z)|e−Λ(dist(z,E)) = 0.

De même, en intégrant par rapport à la mesure de Lebesgue sur le disque D, nous définissonsles espaces BpΛ,E, 1 ≤ p <∞ :

BpΛ,E =f ∈ Hol(D) : ‖f‖pΛ,E,p =

∫

D

|f(z)|pe−pΛ(dist(z,E)) < +∞.

Dans le cas où l’ensemble E = T, nous utilisons les notations B∞Λ , B∞,0

Λ , BpΛ à la place deB∞Λ,E, B∞,0

Λ,E , BpΛ,E. Remarquons que, si limt→0+ Λ(t) <∞, B∞Λ = H∞, B∞,0

Λ = 0, BpΛ = Bp0.Pour une suite de nombres positifs w = wnn telle que | logwn| = o(n), on note

H2w =

f(z) =

∑

n≥0

anzn,

∑

n≥0

|an|2

wn<∞

.

8

Il est bien connu que si la suite wn est log-convexe alors H2w = B2

Λ (il existe Λ telle que

wn ≍(∫ 1

0

r2n+1e−2Λ(1−r) dr)−1

voir [8, Proposition 4.1]).En 1964 Beurling [4] a étudié la cyclicité de S dans l’espace

⋃k≥1H

2wk (muni de la

topologie limite inductive usuelle). Il démontre grâce au théorème d’approximation deBernstein, sous des conditions de régularité sur le poids w, que toute fonction de

⋃k≥1H

2wk

qui ne s’annule pas dans D est cyclique dans⋃k≥1H

2wk si et seulement si S est cyclique

dans⋃k≥1H

2wk si et seulement si

∑

n≥1

logwnn3/2

= ∞.

En 1974 N. Nikolski (voir [39, Section 2.6]) a montré que si lim inft→0Λ(t)

log 1/t> 0, et si la

fonction intérieure S est cyclique dans B∞Λ , alors

∫

0

√Λ(t)

tdt = ∞. (0.4)

Pour la réciproque, il a prouvé que si

la fonction t 7→ tΛ′(t) est strictement croissante, (0.5)

et Λ vérifie la condition (0.4), alors S est cyclique dans B∞Λ . La preuve de ce résultat utilise

un théorème de quasi-analyticité. Pour cela, la condition (0.5) qui traduit en quelque sorteune condition de convexité sur Λ est indispensable dans la démonstration.

Dans la première partie de ce paragraphe nous prouvons les deux résultats suivants :

Théorème 5 Soit Λ une fonction continue, positive et strictement décroissante sur (0, 1].

Alors S(z) = e−1+z1−z est cyclique dans B∞,0

Λ si et seulement si Λ vérifie la condition (0.4).

Théorème 6 Soit 1 ≤ p < ∞, et soit Λ une fonction continue, positive et strictement

décroissante sur (0, 1]. Alors S(z) = e−1+z1−z est cyclique dans BpΛ si et seulement si Λ vérifie

la condition (0.4).

En fait, nous avons pu éliminer les conditions de régularité imposées par N. Nikolski surla fonction Λ en utilisant “la méthode de la résolvante” exposée, par exemple dans [15, 8].Cette technique a été introduite par Carleman et Gelfand ; ensuite Y. Domar l’a utilisépour étudier les idéaux fermés dans certaines algèbres de Banach.

En 1986 Gevorkyan et Shamoyan [20] ont montré, sous quelques conditions de régularitésur la fonction Λ, que la condition

∫

0

Λ(t) dt = ∞, (0.6)

9

est une condition nécessaire et suffisante pour la cyclicité de la fonction S dans l’espaceB∞,0Λ,1. Récemment, El-Fallah, Kellay, et Seip ont amélioré les résultats dû a Beurling et

Nikolski ainsi que le résultat de Gevorkyan et Shamoyan. Leur démonstration est basée surle théorème de la couronne. Pour plus de détails voir [17].

Notons que notre méthode s’applique directement dans le cas où E est un arc fermécontenant le point 1. De plus, pour une fonction Λ suffisamment régulière nous obtenonsdes conditions nécessaires et suffisantes en terme du comportement de E au voisinage dupoint 1.

Dans la deuxième partie de ce paragraphe, nous étudions la cyclicité de S dans B∞,0Λ,E en

fonction de Λ et de E. C’est-à-dire, pour une Λ suffisamment régulière nous obtenons desconditions nécessaires et suffisantes de la cyclicité en fonction de E. Plus précisément, pourune fonction Λ définie par Λ(t) = 1

tw(t)2avec w est une fonction suffisamment régulière,

nous avons le résultat suivant :

Théorème 7 Soit Λ la fonction définie ci-dessus. Alors S est cyclique dans B∞,0Λ,E si et

seulement si l’une des trois quantités suivantes est infinie :∫

eit∈E

dt

|t|w(|t|),

∫

0

dt

|t|w2(|t|),

∑

n

1

w(bn)2log

[1 +

(1−

anbn

)w(bn)

], (0.7)

où (eian , eibn), (e−ibn , e−ian), 0 < an < bn. sont les arcs complémentaires de E dans T.

La preuve est détaillée dans [6, Section 6].Dans le corollaire suivant nous donnons deux exemples où E est un ensemble dénom-

brable

Corollaire 1 Soit Λ la fonction définie ci-dessus. Si E = exp(i · 2−n)n≥1 ∪ 1, alors Sest cyclique dans B∞,0

Λ,E si et seulement si

∑

n≥1

logw(2−n)

w(2−n)2= +∞ ;

si E = exp(i · 2−2n)n≥1 ∪ 1, alors S est cyclique dans B∞,0Λ,E si et seulement si

∫

0

dt

tw(t)2= +∞,

et on retrouve la condition de Gevorkyan–Shamoyan pour E = 1.

Dans la suite, nous donnons encore deux applications du critère général (0.7). Dansla première application, nous obtenons un résultat d’interpolation entre la condition deNikolski et de Gevorkyan–Shamoyan. Pour cela, nous introduisons la condition suivante

∫

0

Λ(t)1−β

tβdt = +∞, 0 ≤ β ≤

1

2. (Cβ)

10

Remarquons que C1/2 est la condition de Nikolski (∫0

√Λ(t)/t dt = +∞) et que C0 est la

condition de Gevorkyan–Shamoyan (∫0Λ(t) dt = +∞).

Pour

Λα(t) =1

t logα(1/t)

on aΛα ∈ (Cβ) ⇐⇒ α(1− β) ≤ 1.

Théorème 8 Soit 0 ≤ β ≤ 1/2, an = exp(−n1−β), n ≥ 1, Eβ = eiann≥1 ∪ 1. La

fonction S(z) = e−1+z1−z est cyclique dans B∞,0

Λα,Eβsi et seulement si Λα ∈ (Cβ).

Le théorème 7 peut aussi être appliqué aux ensembles parfaits comme le montre lethéorème suivant. Nous notons κ la dimension de Hausdorff de l’ensemble triadique deCantor E (voir [18, Section 1.5]), κ = log 2

log 3.

Théorème 9 Soit E un ensemble triadique de Cantor. La fonction S(z) = e−1+z1−z est

cyclique dans B∞,0Λα,E

si et seulement si

α ≤1

1− κ2

.

11

Table des matières

1 Introduction 131.1 Hardy spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1.1 Hp-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.1.2 Cyclic vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.1.3 The Nevanlinna Class . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2 Bergman Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . 161.2.2 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2.3 Cyclicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3 Korenblum spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . 191.3.2 Representation and canonical factorization . . . . . . . . . . . . . . 201.3.3 Cyclicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4 Results and presentation of the work . . . . . . . . . . . . . . . . . . . . . 211.4.1 Cyclic vectors in Korenblum type spaces . . . . . . . . . . . . . . . 211.4.2 Cyclicity in weighted Bergman type spaces . . . . . . . . . . . . . . 25

2 Cyclic vectors in Korenblum type spaces 292.1 Λ-bounded premeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Harmonic functions of restricted growth . . . . . . . . . . . . . . . . . . . 412.3 Cyclic vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3.1 Weights Λ satisfying condition (C1) . . . . . . . . . . . . . . . . . . 462.3.2 Weights Λ satisfying condition (C2) . . . . . . . . . . . . . . . . . . 50

3 Cyclicity in weighted Bergman type spaces 563.1 Generalized Phragmén–Lindelöf Principle . . . . . . . . . . . . . . . . . . . 573.2 Auxiliary estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.3 Proofs of Theorems 3.1 and 3.2 . . . . . . . . . . . . . . . . . . . . . . . . 633.4 An auxiliary domain for general E . . . . . . . . . . . . . . . . . . . . . . 663.5 General E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4 References 75

12

Chapitre 1

Introduction

Throughout this manuscript, we use the following notations : given two functions fand g defined on ∆ we write g ≍ f or g . f if for some 0 < c1 ≤ c2 < +∞ we havec1f ≤ g ≤ c2f , respectively g ≤ c2f on ∆.

Let D be the open unit disk in the complex plane C. Suppose that X is a topologicalvector space of analytic functions on D, with the property that zf ∈ X whenever f ∈ X.Multiplication by z (shift operator) is thus an operator on X, and if X is a Banach space,then it is automatically a bounded operator on the space X. A closed subspace M ⊂ X(Banach space) is said to be invariant (or z-invariant) provided that zM ⊂ M . For afunction f ∈ X, the closed linear span in X of all polynomial multiples of f is a z-invariantsubspace denoted by [f ]X ; it is also the smallest closed z-invariant subspace of X whichcontains f . A function f in X is said to be cyclic (or weakly invertible) in X if [f ]X = X.If the polynomials are dense in X, an equivalent condition is that 1 ∈ [f ]X . For someinformation on cyclic functions see [7] and the references therein.

If the point evaluation functionals,

f 7→ f(z), z ∈ D,

are bounded, then an immediate necessary condition for the function f to be cyclic is thatthe function f have no zeros on D. In general, it is a difficult problem to give necessaryand sufficient conditions for cyclicity.

1.1 Hardy spaces

In this section, we will define the Hardy spaces and the Nevanlinna class, we introducealso some notations and properties of these classes of functions that are going to be neededlater on. The books [19], [16], and [29] are excellent sources of information about Hardyspaces.

13

1.1.1 Hp-Functions

Definitions and factorization.In this section we are concerned with the multiplicative structure of the Hardy spaces, inthat we want to factorize a general Hardy class function as the product of two somewhatsimpler functions, an inner factor and an outer factor.

Let 0 < p <∞ and let f(z) be an analytic function on D. We say that f ∈ Hp := Hp(D)(the Hardy space) if the integrals

∫ 2π

0

|f(reiθ)|pdθ

2π:= ‖f‖pHp ,

are bounded for r < 1. If p = ∞, we say that f ∈ H∞, the algebra of bounded analyticfunction on D if

supz∈D

|f(z)| := ‖f‖∞ <∞.

Suppose that f is a non-zero function of the class H1 on the unit disc. Then f hasnon-tangential limits at almost every point of the unit circle T :

f ∗(eiθ) := limz→eiθ

f(z),

where the limit is taken as z ∈ D approaches the boundary point eiθ within a sector definedby |arg(eiθ − z)| ≤ α, for any constant α < π/2. Furthermore, the function log |f ∗(eiθ)| isLebesgue integrable.

Definition 1.1 An inner function is an H∞(D) function that has unit modulus almosteverywhere on T. An outer function is a function F ∈ H1 which can be written in the form

F (reiθ) = α exp( 1

2π

∫

T

eit + reiθ

eit − reiθφ(eit) dt

), (1.1)

where φ is a real-valued integrable function and |α| = 1.

Proposition 1.2 Let f be an outer function, satisfying (1.1). Then log |f ∗(eiθ)| = φ(eiθ)almost everywhere.

The following theorem gives us an important class of inner functions.

Theorem 1.3 Let S be an inner function without zeros on D, and suppose that S(0)is positive. Then there is a unique positive measure µ, singular with respect to Lebesguemeasure on the unit circle, and a constant α of modulus 1, such that

S(reiθ) := Sµ(reiθ) = α exp

(−

1

2π

∫

T

eit + reiθ

eit − reiθdµ(t)

).

14

These functions S are called singular inner functions. An important example is the atomicsingular inner function

S(z) = exp−

1 + z

1− z

,

with its measure µ concentrated at the point 1.The representation f(z) = eiθS(z)F (z) of function without zeros on D, is unique and

is known as the canonical factorization of a zero-free Hp function.

1.1.2 Cyclic vectors

Theorem 1.4 (A. Beurling) [4].Any closed shift-invariant subspace M of H2 has the form M = θH2, where θ is inner.

We may use Beurling’s theorem to deduce a fairly user-friendly characterization of cyclicfunctions

Corollary 1.5 f ∈ H2 is cyclic if and only if it is an outer function.

1.1.3 The Nevanlinna Class

An analytic function f in D is said to be in the Nevanlinna class, f ∈ N if the integrals

∫ 2π

0

log+ |f(reiθ)|dθ

2π,

are bounded for r < 1, where log+ a = maxlog a, 0. It is clear that, Hp ⊂ N for allp > 0, because log+ |f(z)| ≤ 1

p|f(z)|p. It is known that functions in the Nevanlinna class

are quotients of H∞ functions, so each function f ∈ N has a non-tangential limit f ∗(eiθ)at almost every boundary point eiθ. Furthermore, given a zero-free function f ∈ N , thereexists a unique outer function F and unique singular inner functions Sµ(z) and Sν(z) withmutually singular associated measures such that (see [16, Thm. 2.9, p. 25])

f(z) =Sµ(z)

Sν(z)F (z).

1.2 Bergman Spaces

in this section, we introduce some properties of Bergman spaces, and we will reviewsome results and problems concerning the cyclicity problem in these spaces.

15

1.2.1 Definitions and properties

For 0 < p < ∞, the Bergman space Ap(D) := Ap of the disc is the space of analyticfunctions in D for which

‖f‖pp := ‖f‖pAp =1

π

∫ ∫

D

|f(z)|p dxdy, (z = x+ iy).

Equipped with the above norm, Ap is a Banach space for 1 ≤ p < ∞. If 0 < p < 1, thetriangle inequality is replaced by the inequality ‖f + g‖pp ≤ ‖f‖pp + ‖g‖pp (a quasi-Banachspace for 0 < p < 1). In particular, Ap is always a linear metric space.

The following theorem asserts that functions in a Bergman space cannot grow toorapidly near the boundary.

Theorem 1.6 The point evaluation is a bounded linear functional in each Bergman spaceAp. More specifically, every function f in Ap satisfies the following estimate :

|f(z)| ≤ ‖f‖pπ− 1

p1

(1− |z|)2p

, z ∈ D.

A2 is a reproducing kernel Hilbert space with respect to the inner product

〈f, g〉 =

∫ ∫

D

f(z)g(z) dxdy,

and its kernel is called the Bergman kernel, and denoted by K(w, z).

Lemma 1.7 The Bergman kernel for the open unit disc D is given by

K(w, z) =1

π

1

(1− wz)2, for w, z ∈ D.

Note also that the Bergman space Ap contains Hp as a dense subspace, and we havef ∗(eiθ) ∈ Lp(T), f ∈ Hp (non-tangential limits exists at almost every boundary point forHp functions). This is however not the case for Ap(D). In fact, there is a function in itwhich fails to have radial limits at every point of T. For example, we can take the function

f(z) =+∞∑n=1

z22n

∈ Ap \N ( see [12]) ; then f(z) has no finite radial limits.

1.2.2 Factorization

We pass to the factorization theory in the Bergman space. Let M be a proper closedsubspace of Ap invariant under multiplication by z. Denote by n the smallest nonnegativeinteger such that there exists a function f ∈M with f (n)(0) 6= 0, and consider the followingextremal problem

supRef (n)(0) : f ∈M, ‖f‖Ap = 1

.

16

Suppose that M is a cyclic invariant subspace (singly generated invariant subspace), andthat the generator is g ∈ Ap. Then the above problem has a unique solution, which wedenote by G and call the extremal solution (or canonical divisor) for the subspace M . Itis proved in [2] (see also [24, Chapter 3]) that M = [G], and that G have the so-calledexpansive multiplier property, or equivalently, the contractive divisibility property on M :

∥∥∥ gG

∥∥∥Ap

≤ ‖g‖Ap , g ∈M.

Theorem 1.8 Suppose that 0 < p < ∞ and f ∈ Ap. Then there exists G (the solutionto the extremal problem for M = [f ]), and a cyclic vector F ∈ Ap such that f = GF .Furthermore,

‖F‖Ap ≤ ‖f‖Ap .

In the classical theory of Hardy spaces Hp, the inner-outer factorization is unique (upto a unimodular constant multiple of the inner factor). Unfortunately, the factorization inBergman spaces Ap does not have such a strong uniqueness property (for the explication,see [24, Corollary 8.8]).

1.2.3 Cyclicity

Cyclic Vectors as Outer functions.Boris Korenblum introduced in [34] the notion of outer function for Bergman spaces Ap interms of domination and proved that a cyclic function in Ap necessarily is outer. We writef ≻ g (or g ≺ f) and say that f dominates g in Ap if

‖fh‖Ap ≥ ‖gh‖Ap , h ∈ H∞.

Note that for Hp-functions f ≻ g is equivalent to |f(eiθ)| ≥ |g(eiθ)| almost everywhere.The following definition is motivated by the properties of (Beurling) outer function.

Definition 1.9 We say that a function f ∈ Ap is Ap-outer if |f(0)| ≥ |g(0)| wheneverg ≺ f in Ap .

The following theorem was conjectured by Korenblum in [34], and later proved by Aleman,Richter, and Sundberg in [2].

Theorem 1.10 Let f ∈ Ap with 0 < p < ∞. Then f is Ap-outer if and only if it is cyclicin Ap.

Singular inner functions are sometimes cyclic.A zero-free function f in the class Ap ∩N (f(z) = Sµ(z)

Sν(z)F (z)) is cyclic in Ap if and only if

its singular inner function Sµ is cyclic,

Sµ(z) = exp[−

1

2π

∫ 2π

0

eiθ + z

eiθ − zdµ(θ)

], z ∈ D,

17

i.e. if and only if its associated positive singular measure µ places no mass on any Carlesonset (Beurling-Carleson sets). Carleson sets constitute a class of thin subsets of T, they willbe discussed below. The necessity of this Carleson set condition was proved by H. S. Shapiroin 1967 [42, Theorem 2], and the sufficiency was proved independently and using differentarguments by B. Korenblum in 1977 [33] and J. Roberts in 1979 [40, Theorem 2].

Furthermore, H. S. Shapiro [43, Theorem 2] showed that there exists an absoluteconstant C > 0 such that

|Sµ(z)| ≥ exp(−Cωµ(1− |z|)

1− |z|

),

where ωµ is the modulus of continuity of µ defined by

ωµ(t) = supµ(I) : |I| < t, I a sub-arc of T.

In particular, the singular inner functions with ωµ(t) = O(t log(1

t))

are cyclic in Bergmanspaces Ap.Invertibility and cyclicity.A function f in a space X of analytic functions is said to be invertible if 1/f also belongsto X. In the classical theory of Hardy spaces, it is known that every invertible function inHp is necessarily cyclic in Hp. This is also true in the A−∞ space (the Korenblum space,see definition 1.12 below) ; an invertible function f in A−∞,

|f(z)| ≥ C(1− |z|)c z ∈ D, (1.2)

for some positive numbers C, c, is always cyclic in A−∞.In the case of Bergman spaces, H. S. Shapiro (see [42]), posed the following question.

Suppose that f satisfies (1.2). Is f cyclic in A2 ? This question was settled in the negativeby A. Borichev and H. Hedenmalm [7]. The construction involved first finding harmonicfunctions with special growth properties, and then forming the zero-free function obtainedby harmonic conjugation plus exponentiation.

The following theorem due to H. S. Shapiro [42, 44], gives a slightly more restrictivesufficient condition for cyclicity

Theorem 1.11 If f ∈ Aq for some q > p, and if there are positive constants c and C suchthat |f(z)| ≥ C(1− |z|)c for all z ∈ D, then f is cyclic in Ap.

So, it is natural to ask (see for example, Question 6 in [45], posed by A. L. Shields)if there exists any decreasing radial function φ(z) = φ(|z|), with φ(r) → 0, as r → 1−

such that the conditions f ∈ Ap and |f(z)| ≥ φ(|z|) for z ∈ D would imply that f iscyclic in Ap. In [10] A. Borichev shows that the answer is affirmative. More specifically, thefunction

φ(t) = exp

[−

(log

1

1− t

)1/(2+α)]

α > 0,

has the desired property.It remains an open problem to give an explicit characterization of the cyclic functions

in Bergman spaces.

18

1.3 Korenblum spaces

In this section we will give a short introduction to Korenblum’s work on generalizedNevanlinna theory. Furthermore, we review some results on the cyclicity problem in thesespaces.

1.3.1 Definitions and properties

The classical representation and factorization theory due to R. Nevanlinna [38] is ba-sed on the application of the Poisson-Jensen formula to smaller disks |z| ≤ r < 1 andon a subsequent transition to the limit as r → 1 involving an application of the Riesz-Herglotz formula. The Riesz-Herglotz theorem does not directly applies to general classesof harmonic functions which makes difficult to extend Nevanlinna theory to these classes.

An important contribution was made here by Korenblum who introduced what is knownow as "Korenblum spaces"

Definition 1.12 The Korenblum space A−∞ is a topological algebra of analytic functionsf in D that satisfy the following estimate

|f(z)| ≤ Cf1

(1− |z|)c, Cf , c > 0,

i.e

A−∞ = ∪c>0A−c =

⋃

c>0

f ∈ Hol(D) : |f(z)| ≤ Cf

1

(1− |z|)c

.

Note that the class A−∞ is the smallest extension ring of algebra H∞ invariant underdifferentiation. Furthermore, A−∞ contains (in fact, is the union of) the Bergman spacesAp (0 < p <∞). Since the set of polynomials is dense in A−∞ it is readily seen that everyinvariant subspace for the operator Mz (multiplication by z) is a closed ideal in the algebraA−∞, and vice versa.

With the norm‖f‖A−c = sup

z∈D(1− |z|)c|f(z)| <∞,

A−c becomes a Banach space and for every c2 ≥ c1 > 0, the inclusion A−c1 → A−c2 iscontinuous. The topology on

A−∞ = ∪c>0A−c,

is the locally-convex inductive limit topology, i.e. each of the inclusions A−c → A−∞ iscontinuous and the topology is the largest locally-convex topology with this property. Asequence fnn ∈ A−∞ converges to f ∈ A−∞ if and only if there exists N > 0 such thatall fn and f belong to A−N , and limn→+∞ ‖fn − f‖A−N = 0.

19

1.3.2 Representation and canonical factorization

Let B(T) be the set of all (open, half-open and closed) arcs of T including all the singlepoints and the empty set. The elements of B(T) will be called intervals.

Definition 1.13 A real valued function defined on B(T) is called a premeasure if the fol-lowing conditions hold :

1. µ(T) = 0

2. µ(I1 ∪ I2) = µ(I1) + µ(I2) for every I1, I2 ∈ B(T) such that I1 ∩ I2 = ∅ andI1 ∪ I2 ∈ B(T)

3. limn→+∞

µ(In) = 0 for every sequence of embedded intervals, In+1 ⊂ In, n ≥ 1, such that⋂n In = ∅.

Definition 1.14 A premeasure µ is called κ-bounded if there exists a positive constant Csuch that

µ(I) ≤ C|I| log2πe

|I|, for all I ∈ B(T).

The set of all κ-bounded premeasures will be denoted B+κ .

Theorem 1.15 [31] Let h be a real-valued harmonic function on the unit disk such thath(0) = 0 and

h(z) = O(| log(1− |z|)|), |z| → 1, z ∈ D.

Then the following statements hold.

1. For every open arc I of the unit circle T the following limit exists :

µ(I) = limr→1−


∫

I

h(rξ) |dξ| <∞.

2. µ is a κ-bounded premeasure.

3. The function h is the Poisson integral of the premeasure µ :

h(z) =

∫ 2π

0

1− |z|2

|eiθ − z|2dµ(θ), z ∈ D;

to define this integral we integrate by parts.

Let f be a zero-free function in A−∞ such that f(0) = 1. According to Theorem 1.15,there is a premeasure µf ∈ B+

κ such that

f(z) = exp

∫ 2π

0

eiθ + z

eiθ − zdµf (θ).

20

1.3.3 Cyclicity

Every κ-bounded premeasure µ generates a finite Borel measure on Carleson sets, i.e.on closed subsets of T which have measure zero, and whose complementary arcs In on T

satisfy the relation ∑

n

|In| log2πe

|In|<∞.

This Borel measure (defined on B : the set of all Borel sets B ⊂ T such that B is Carlesonset) is called the κ-singular part of κ-bounded premeasure µ and denoted by µs.

A description of all closed ideals (invariant subspaces for the operator of multiplicationby z) of the topological algebra A−∞ is given in [32]. Each such zero-free closed ideal (zero-free z-invariant subspace) is uniquely characterized by its κ-singular measure which iscompletely analogous to the case of the invariant subspaces of H2 described in the classicaltheory of Beurling [4].

Theorem 1.16 [32] An element f in Korenblum space A−∞ is cyclic if and only if

1. f(z) has no zeros in D.

2. The κ-singular measure associated with f is zero.

Theorem 1.17 [24, Theorem 7.3] Let 0 < p <∞, and let f ∈ Aq, q > p. Then f is cyclicin Ap if and only if f has no zeros in D and the κ-singular measure associated with f iszero.

Hopefully, new information on the structure of the closed ideals in A−∞ would mark atthe same time some progress in the cyclicity problem for Bergman spaces Ap.

For other applications of the premeasures technique, one should mention a paper byN. G. Makarov in [37] describing z-invariant subspaces of the space C∞ (the space ofinfinitely differentiable functions on the unit circle T), and a paper by B. Korenblum [35]on the zero sets in Ap, A−p spaces.

1.4 Results and presentation of the work

1.4.1 Cyclic vectors in Korenblum type spaces

In this section we extend the results of two papers by Korenblum [31, 32] on Λ-boundedpremeasures (see also [24, Chapter 7]) from the case Λ(t) = log(1/t) to the general case.

In the following a majorant Λ will always denote a positive non-increasing convexdifferentiable function on (0,1] such that :

1. Λ(0) = +∞

2. tΛ(t) is a continuous, non-decreasing and concave function on [0, 1], and tΛ(t) → 0as t→ 0.

3. There exists α ∈ (0, 1) such that tαΛ(t) is non-decreasing.

21

4. For some C > 0, every t ∈ (0, 1) we have

Λ(t2) ≤ CΛ(t). (1.3)

Typical examples of majorants Λ are log+ log+(1/x), (log(1/x))p, p > 0.Denote by A−∞

Λ the Korenblum type space associated with the majorant Λ, defined by

A−∞Λ = ∪c>0A

−cΛ =

⋃

c>0

f ∈ Hol(D) : |f(z)| ≤ exp(cΛ(1− |z|))

.

Note that the set of all polynomials is dense in the topological algebra (with respect topointwise multiplication and the natural injective limit topology) A−∞

Λ . Furthermore, Mz

and the point evaluation functionals are bounded on A−∞Λ .

Definition 1.18 A premeasure µ is said to be Λ-bounded, if there is a positive number Cµsuch that

µ(I) ≤ Cµ|I|Λ(|I|) (1.4)

for any interval I.

The minimal number Cµ is called the norm of µ and is denoted by ‖µ‖+Λ ; the set of allreal premeasures µ such that ‖µ‖+Λ < +∞ is denoted by B+

Λ .Given a closed non-empty subset F of the unit circle T, we define its Λ-entropy as

follows :EntrΛ(F ) =

∑

n

|In|Λ(|In|),

where Inn are the component arcs of T \ F , and |I| denotes the normalized Lebesguemeasure of I on T. We set EntrΛ(∅) = 0.

We say that a closed set F is a Λ-Carleson set if F is non-empty, has Lebesgue measurezero (i.e |F | = 0), and EntrΛ(F ) < +∞.

Denote by CΛ the set of all Λ-Carleson sets and by BΛ the set of all Borel sets B ⊂ T

such that B ∈ CΛ.We can now introduce the notion of Λ-singular measures.

Definition 1.19 A function σ : BΛ → R is called a Λ-singular measure if

1. σ is a finite Borel measure on every set in CΛ (i.e. σ∣∣F is a Borel measure on T).

2. There is a constant C > 0 such that

|σ(F )| ≤ CEntrΛ(F )

for all F ∈ CΛ.

22

Given a premeasure µ in B+Λ , its Λ-singular part is defined by :

µs(F ) = −∑

n

µ(In), (1.5)

where F ∈ CΛ and Inn is the collection of complementary intervals to F in T. Using theargument in [31, Theorem 6] one can see that µs extends to a Λ-singular measure on BΛ.

Next we introduce the notion of Λ-absolutely continuous premeasure, which will giveus a cyclicity criterion.

Definition 1.20 A premeasure µ in B+Λ is said to be Λ-absolutely continuous if there exists

a sequence of Λ-bounded premeasures (µn)n such that :

1. supn ‖µn‖+Λ < +∞.

2. supI∈B(T) |(µ+ µn)(I)| → 0 as n→ +∞.

The proof of the next result follows an argument by Korenblum in [32].

Theorem 1.21 Let µ be a premeasure in B+Λ . Then µ is Λ-absolutely continuous if and

only if its Λ-singular part µs is zero.

The only if part holds in a more general situation considered by Korenblum, [36, Co-rollary, p.544]. On the other hand, the if part does not hold for differences of Λ-boundedpremeasures (premeasures of Λ-bounded variation), see [36, Remark, p.544].

Harmonic functions of restricted growth and premeasures.Every bounded harmonic function can be represented via the Poisson integral of its boun-dary values. In the following theorem we show that a large class of real-valued harmonicfunctions in the unit disk D can be represented as the Poisson integrals of Λ-boundedpremeasures.

The following theorem is stated by Korenblum in [36, Theorem 1, p. 543] without proof,in a more general situation.

Theorem 1.22 Let h be a real-valued harmonic function on the unit disk such that h(0) =0 and

h(z) = O(Λ(1− |z|)), |z| → 1, z ∈ D.



µ(I) = limr→1−


∫

I

h(rξ) |dξ| <∞.

2. µ is a Λ-bounded premeasure.


h(z) =

∫ 2π

0

1− |z|2

|eiθ − z|2dµ(θ), z ∈ D.

23

Conversely, every Λ-bounded premeasure µ generates a harmonic function h(z) in D (thePoisson integral of µ) such that

h(z) = O(Λ(1− |z|)), |z| → 1, z ∈ D, (1.6)

by the formula

h(z) =

∫

T

1− |z|2

|eiθ − z|2dµ.

Let f be a zero-free function in A−∞Λ such that f(0) = 1. According to Theorem 1.22,

there is a premeasure µf := µ ∈ B+Λ such that

f(z) = fµ(z) := exp

∫ 2π

0

eiθ + z

eiθ − zdµ(θ). (1.7)

The following result follows immediately from Theorem 1.21.

Theorem 1.23 Let f ∈ A−∞Λ be a zero-free function such that f(0) = 1. If (µf )s ≡ 0,

then f is cyclic in A−∞Λ .

From now on, we deal with the statements converse to Theorem 1.23. We establish tworesults valid for different growth ranges of the majorant Λ. More precisely, we consider thefollowing growth and regularity assumptions :

for every c > 0, the function x 7→ exp[cΛ(1/x)

]is concave for large x, (C1)

limt→0

Λ(t)

log(1/t)= ∞. (C2)

Examples of majorants Λ satisfying condition (C1) include



(log(1/x))p, p > 1.

Thus, we consider majorants which grow less rapidly than the Korenblum majorant (Λ(x) =log(1/x)) in Case 1 or more rapidly than the Korenblum majorant in Case 2.

Theorem 1.24 Let µ ∈ B+Λ , and let the majorant Λ satisfy condition (C1). Then the

function fµ is cyclic in A−∞Λ if and only if µs ≡ 0.



24

Theorems 1.24 and 1.25 together give a positive answer to a conjecture by Deninger [14,Conjecture 42].

Now we give two examples that show how the cyclicity property of a fixed functionchanges in a scale of A−∞

Λ spaces.

Example 1.26 Let Λα(x) = (log(1/x))α, 0 < α < 1, and let 0 < α0 < 1. There exists asingular inner function Sµ such that

Sµ is cyclic in A−∞Λα

⇐⇒ α > α0.

Example 1.27 Let Λα(x) = (log(1/x))α, 0 < α < 1, and let 0 < α0 < 1. There exists apremeasure µ such that µs is infinite,

fµ is cyclic in A−∞Λα

⇐⇒ α > α0,

where fµ is defined by (1.7).

It looks like the subspaces [fµ]A−∞Λα

, α ≤ α0, contain no nonzero Nevanlinna class func-tions. For a detailed discussion on Nevanlinna class generated invariant subspaces in theBergman space (and in the Korenblum space) see [23, Chapter 6].

1.4.2 Cyclicity in weighted Bergman type spaces

In this part, we investigate the question of weak invertibility (cyclicity) of the atomicsingular inner function S(z) = e−

1+z1−z with its measure µ concentrated at one point, in the

weighted Bergman type spaces.Given a positive non-increasing continuous function Λ on (0, 1] and E ⊂ T = ∂D, we

denote by B∞Λ,E the space of all analytic functions f on D such that


|f(z)|e−Λ(dist(z,E)) < +∞,

and by B∞,0Λ,E its separable subspace

B∞,0Λ,E =

f ∈ B∞



Analogously, integrating with respect to area measure on the disc, we define the spacesBpΛ,E, 1 ≤ p <∞ :


∫

D


If E = T, we use the notations B∞Λ , B∞,0

Λ , BpΛ. Let us remark that either Λ(0+) = +∞or B∞

Λ = H∞, B∞,0Λ = 0, BpΛ = Bp0.

25

Given a sequence of positive numbers w = (wn) such that | logwn| = o(n), n→ ∞, wedefine by H2

w the space of functions f analytic in the unit disc and such that

f(z) =∑

n≥0

anzn,

∑

n≥0

|an|2

wn<∞.

It is known that for log-convex sequences w such spaces coincide with the B2Λ,T spaces for

Λ defined by w (see [8]).The cyclicity question in the weighted Bergman type spaces BpΛ, 1 ≤ p <∞, goes back

to Carleman and Keldysh. In particular, Keldysh [25] proved in 1945 that the singularinner function with one point singular mass S(z) = e−

1+z1−z is not cyclic in B2

0.In 1964 Beurling [5] studied the cyclicity of the function S in space

⋃k≥1H

2wk , equip-

ped with the natural topology of an inductive limit (this space is a topological algebrawith respect to ordinary multiplication of functions). Using the Bernstein’s approximationtheorem he produced under mild restrictions on the regularity of growth, a necessary andsufficient condition for the localization of all principal ideals (in other words, a necessaryand sufficient condition for each function in

⋃k≥1H

2wk lacking zeros in D to be cyclic in⋃

k≥1H2wk). This condition is the divergence of the series

∑

n≥1

logwnn3/2

.

In 1974 Nikolski [39, Section 2.6] proved that if lim inft→0Λ(t)

log 1/t> 0, and S is cyclic in B∞

Λ ,then ∫

0

√Λ(t)

tdt = ∞. (1.8)

In the opposite direction, he proved that if

the function t 7→ tΛ′(t) does not decrease, (1.9)

and (1.8) holds, then S is cyclic in B∞Λ . Since the proof relied on the quasianalyticity pro-

perty of an auxiliary class of functions, this convexity type condition (1.9) was indispensablehere.

In the first part of this subsection we prove the following results :

Theorem 1.28 Let Λ be a positive non-increasing continuous function on (0, 1]. Then

S(z) = e−1+z1−z is cyclic in B∞,0

Λ if and only if Λ satisfies (1.8).

Theorem 1.29 Let 1 ≤ p <∞ and let Λ be a positive non-increasing continuous function

on (0, 1]. Then S(z) = e−1+z1−z is cyclic in BpΛ if and only if Λ satisfies (1.8).

Thus, we are able to get rid of any regularity condition on Λ. That is possible becausewe use the so-called resolvent transform method exposed, for example, in [15, 8]. This

26

technique was introduced by Carleman and Gelfand ; it was later rediscovered and usedupon by Domar to study closed ideals in Banach algebras.

In 1986 Gevorkyan and Shamoyan [20] obtained a necessary and sufficient condition,∫

0

Λ(t) dt = ∞, (1.10)

for cyclicity of S in B∞,0Λ,1 under some regularity conditions on Λ. Recently, El-Fallah,

Kellay, and Seip [17] improved the results of Beurling and Nikolski for cyclicity of boundedzero-free functions in H2

w spaces. Furthermore, improving on the result by Gevorkyan–Shamoyan they obtained that (1.10) is necessary and sufficient for cyclicity of S in B∞,0

Λ,1,the only regularity condition being that Λ is decreasing. Their method of proof (applyingthe Corona theorem) is quite different from what we use here.

It is now a natural question to describe Λ such that S is cyclic in B∞,0Λ,E , B2

Λ,E in termsof the behavior of E near the point 1. Our method applies directly in the case where Eis a closed arc containing 1. Furthermore, for sufficiently regular Λ we get necessary andsufficient conditions in terms of E.

In the second part of this subsection, for sufficiently regular Λ we get necessary andsufficient conditions in terms of E. More precisely, for Λ defined by Λ(t) = 1

tw(t)2with

sufficiently regular w, the function S is cyclic in B∞,0Λ,E if and only if one of the following

three quantities is infinite :∫

eit∈E

dt

|t|w(|t|),

∫

0

dt

|t|w2(|t|),

∑

n

1

w(bn)2log

[1 +

(1−

anbn

)w(bn)

], (1.11)

where the sums runs by all the complementary arcs to E : (eian , eibn), (e−ibn , e−ian), 0 <an < bn.

Two simple examples of the set E are considered in the following corollary

Corollary 1.30 Let Λ be the function described above. If E = exp(i · 2−n)n≥1 ∪ 1,then S is cyclic in B∞,0

Λ,E if and only if

∑

n≥1

logw(2−n)

w(2−n)2= +∞ ;

if E = exp(i · 2−2n)n≥1 ∪ 1, then S is cyclic in B∞,0Λ,E if and only if

∫

0

dt

tw(t)2= +∞,

and we return to the Gevorkyan–Shamoyan condition valid for E = 1.

Next we give two more applications of the general criterion (1.11).

27

First, we get a result interpolating between the theorems of Nikolski and Gevorkyan–Shamoyan. Let us introduce a condition

∫

0

Λ(t)1−β

tβdt = +∞, 0 ≤ β ≤

1

2(Cβ)

(compare to those by Nikolski (∫0

√Λ(t)/t dt = +∞, β = 1/2) and by Gevorkyan–

Shamoyan (∫0Λ(t) dt = +∞, β = 0)).

For

Λα(t) =1

t logα(1/t)

we haveΛα ∈ (Cβ) ⇐⇒ α(1− β) ≤ 1.

Theorem 1.31 Let 0 ≤ β ≤ 1/2, an = exp(−n1−β), n ≥ 1, Eβ = eiann≥1 ∪ 1. The

function S(z) = e−1+z1−z is cyclic in B∞,0

Λα,Eβif and only if Λα ∈ (Cβ).

We also give another application of our result, in the case when E is the Cantor ternaryset. Denote by κ the Hausdorff dimension of E (see [18, Section 1.5]), κ = log 2

log 3.

Theorem 1.32 Let E be the Cantor ternary set. The function S(z) = e−1+z1−z is cyclic in

B∞,0Λα,E

if and only if

α ≤1

1− κ2

.

28

Chapitre 2

Cyclic vectors in Korenblum type spaces

In this chapter, a majorant Λ will always denote a positive non-increasing convex dif-ferentiable function on (0,1] such that :

1. Λ(0) = +∞

2. tΛ(t) is a continuous, non-decreasing and concave function on [0, 1], and tΛ(t) → 0as t→ 0.

3. There exists α ∈ (0, 1) such that tαΛ(t) is non-decreasing.

4. There exists C > 0, such thatΛ(t2) ≤ CΛ(t). (2.1)

Typical examples of majorants Λ are log+ log+(1/x), (log(1/x))p, p > 0.Furthermore, we shall be interested mainly in studying cyclic vectors in the space

A−∞Λ , by generalized the theory of premeasures introduced by Korenblum ; here A−∞

Λ isthe Korenblum type space associated with the majorant Λ, defined by

A−∞Λ = ∪c>0A

−cΛ =

⋃

c>0

f ∈ Hol(D) : |f(z)| ≤ exp(cΛ(1− r))

.

2.1 Λ-bounded premeasures

In this section we extend the results of two papers by Korenblum [31, 32] on Λ-boundedpremeasures (see also [24, Chapter 7]) from the case Λ(t) = log(1/t) to the general case.

Let B(T) be the set of all (open, half-open and closed) arcs of T including all the singlepoints and the empty set. The elements of B(T) will be called intervals.

Definition 2.1 A real function defined on B(T) is called a premeasure if the followingconditions hold :

1. µ(T) = 0

2. µ(I1 ∪ I2) = µ(I1) + µ(I2) for every I1, I2 ∈ B(T) such that I1 ∩ I2 = ∅ andI1 ∪ I2 ∈ B(T)

29

3. limn→+∞

µ(In) = 0 for every sequence of embedded intervals, In+1 ⊂ In, n ≥ 1, such that⋂n In = ∅.

Given a premeasure µ, we introduce a real valued functionfµ on (0, 2π] defined as

follows :fµ(θ) = µ(Iθ),

whereIθ =

ξ ∈ T : 0 ≤ arg ξ < θ

.

The functionfµ satisfies the following properties :

(a)fµ(θ−) exists for every θ ∈ (0,2π] and

fµ(θ+) exists for every θ ∈ [0, 2π)

(b)fµ(θ) = limt→θ−

fµ(t) for all θ ∈ (0,2π]

(c)fµ(2π) = limθ→0+

fµ(θ) = 0.

Furthermore, the functionfµ(θ) has at most countably many points of discontinuity.

Definition 2.2 A real premeasure µ is said to be Λ-bounded, if there is a positive numberCµ such that

µ(I) ≤ Cµ|I|Λ(|I|) (2.2)

for any interval I.

The minimal number Cµ is called the norm of µ and is denoted by ‖µ‖+Λ ; the set of allreal premeasures µ such that ‖µ‖+Λ < +∞ is denoted by B+

Λ .

Definition 2.3 A sequence of premeasures µnn is said to be Λ-weakly convergent to apremeasure µ if :

1. supn ‖µn‖+Λ < +∞, and

2. for every point θ of continuity offµ we have limn→∞

fµn(θ) =

fµ(θ).

In this situation, the limit premeasure µ is Λ-bounded.

Given a closed non-empty subset F of the unit circle T, we define its Λ-entropy asfollows :

EntrΛ(F ) =∑

n

|In|Λ(|In|),

where Inn are the component arcs of T \ F , and |I| denotes the normalized Lebesguemeasure of I on T. We set EntrΛ(∅) = 0.

We say that a closed set F is a Λ-Carleson set if F is non-empty, has Lebesgue measurezero (i.e |F | = 0), and EntrΛ(F ) < +∞.

Denote by CΛ the set of all Λ-Carleson sets and by BΛ the set of all Borel sets B ⊂ T

such that B ∈ CΛ.

30

Definition 2.4 A function σ : BΛ → R is called a Λ-singular measure if

1. σ is a finite Borel measure on every set in CΛ (i.e. σ∣∣F is a Borel measure on T).

2. There is a constant C > 0 such that

|σ(F )| ≤ CEntrΛ(F )

for all F ∈ CΛ.

Given a premeasure µ in B+Λ , its Λ-singular part is defined by :

µs(F ) = −∑

n

µ(In), (2.3)

where F ∈ CΛ and Inn is the collection of complementary intervals to F in T. Using theargument in [31, Theorem 6] one can see that µs extends to a Λ-singular measure on BΛ.

Proposition 2.5 If µ is a Λ-bounded premeasure, F ∈ CΛ, then µs∣∣F is finite and non-

positive.

Proof. Let F ∈ CΛ. We are to prove that µs(F ) ≤ 0.Let Inn be the (possibly finite) sequence of the intervals complementary to F in T.

For N ≥ 1, we consider the disjoint intervals JNn 1≤n≤N such that T \⋃Nn=1 In =

⋃Nn J

Nn .

Then

−N∑

n=1

µ(In) =N∑

n=1

µ(JNn ) ≤ ‖µ‖+Λ

N∑

n=1

|JNn |Λ(|JNn |).

Furthermore, each interval JNn is covered by intervals Im ⊂ JNn up to a set of measure zero,and max1≤n≤N |JNn | → 0 as N → ∞ (If the sequence Inn is finite, then all JNn are singlepoints for the corresponding N). Therefore,

−N∑

n=1

µ(In) ≤ ‖µ‖+Λ

N∑

n=1

∑

Im⊂JNn

|Im|Λ(|Im|) ≤ ‖µ‖+Λ∑

n>N

|In|Λ(|In|).

Since F is a Λ-Carleson set,

− limN→∞

N∑

n=1

µ(In) ≤ 0.

Thus, µs∣∣F ≤ 0.

Given a closed subset F of T, we denote by F δ its δ-neighborhood :

F δ = ζ ∈ T : d(ζ, F ) ≤ δ.

31

Proposition 2.6 Let µ be a Λ-bounded premeasure and let µs be its Λ-singular part. Thenfor every F ∈ CΛ we have

µs(F ) = limδ→0

µ(F δ). (2.4)

Proof. Let F ∈ CΛ, and let Inn, |I1| ≥ |I2| ≥ . . . , be the intervals of the complementto F in T. We set

I(δ)n =eiθ : dist(eiθ,T \ In) > δ

.

Then for |In| ≥ 2δ, we haveIn = I1n ⊔ I

(δ)n ⊔ I2n

with |I1n| = |I2n| = δ. We see that

µ(F δ) = −∑

|In|>2δ

µ(I(δ)n ).

Using relation (2.3) we obtain that

−µs(F ) =∑

n

µ(In)

=∑

|In|≤2δ

µ(In) +∑

|In|>2δ

[µ(I1n) + µ(I(δ)n ) + µ(I2n)

]

=∑

|In|≤2δ

µ(In)− µ(F δ) +∑

|In|>2δ

[µ(I1n) + µ(I2n)

].

Therefore,µ(F δ)− µs(F ) =

∑

|In|≤2δ

µ(In) +∑

|In|>2δ

[µ(I1n) + µ(I2n)

]

The first sum tends to zero as δ → 0, and it remains to prove that

limδ→0

∑

|In|>2δ

µ(I1n) = 0. (2.5)

We have

∑

|In|>2δ

µ(I1n) ≤ C∑

|In|>δ

δΛ(δ) = C∑

|In|>δ

δΛ(δ)

|In|Λ(|In|)· |In|Λ(|In|).

Since the function t 7→ tΛ(t) does not decrease, we have

δΛ(δ)

|In|Λ(|In|)≤ 1, |In| > δ.

Furthermore,

limδ→0

δΛ(δ)

|In|Λ(|In|)= 0, n ≥ 1.

32

Since ∑

n≥1

|In|Λ(|In|) <∞,

we conclude that (2.5), and, hence, (2.4) hold.

Definition 2.7 A premeasure µ in B+Λ is said to be Λ-absolutely continuous if there exists

a sequence of Λ-bounded premeasures (µn)n such that :

1. supn ‖µn‖+Λ < +∞.

2. supI∈B(T) |(µ+ µn)(I)| → 0 as n→ +∞.

Theorem 2.8 Let µ be a premeasure in B+Λ . Then µ is Λ-absolutely continuous if and

only if its Λ-singular part µs is zero.

The only if part holds in a more general situation considered by Korenblum, [36, Co-rollary, p.544]. On the other hand, the if part does not hold for differences of Λ-boundedpremeasures (premeasures of Λ-bounded variation), see [36, Remark, p.544].

To prove this theorem we need several lemmas. The first one is a linear programminglemma from [24, Chapter 7].

Lemma 2.9 Consider the following system of N(N+1)/2 linear inequalities in N variablesx1, . . . , xN

l∑

j=k

xj ≤ bk,l, 1 ≤ k ≤ l ≤ N,

subject to the constraint : x1 + x2 + · · ·+ xN = 0. This system has a solution if and only if∑

n

bkn,ln ≥ 0

for every simple covering P = [kn, ln]n of [1, N ].

The following lemma gives a necessary and sufficient conditions for a premeasure in B+Λ

to be Λ-absolutely continuous.

Lemma 2.10 Let µ be a Λ-bounded premeasure. Then µ is Λ-absolutely continuous if andonly if there is a positive constant C > 0 such that for every ε > 0 there exists a positiveM such that the system

xk,l ≤ M |Ik,l|Λ(|Ik,l|)µ(Ik,l) + xk,l ≤ minC|Ik,l|Λ(|Ik,l|), ε

xk,l =l−1∑s=k

xs,s+1

x0,N = 0

(2.6)

33

in variables xk,l, 0 ≤ k < l ≤ N , has a solution for every positive integer N . Here Ik,l arethe half-open arcs of T defined by

Ik,l =eiθ : 2π

k

N≤ θ < 2π

l

N

.

Proof.Suppose that µ is Λ-absolutely continuous and denote by µn a sequence of Λ-bounded

premeasures satisfying the conditions of Definition 2.7. Set

C = supn

‖µ+ µn‖+Λ , M = sup

n‖µn‖

+Λ ,

and let ε > 0. For large n, the numbers xk,l = µn(Ik,l), 0 ≤ k < l ≤ N , satisfy relations(2.6) for all N .

Conversely, suppose that for some C > 0 and for every ε > 0 there existsM =M(ε) > 0such that for every N there are xk,lk,l (depending on N) satisfying relations (2.6). Weconsider the measures dµN defined on Is,s+1, 0 ≤ s < N , by

dµN(ξ) =xs,s+1

|Is,s+1||dξ|,

where |dξ| is normalized Lebesgue measure on the unit circle T. To show that µN ∈ B+Λ ,

it suffices to verify that the quantity supIµ(I)

|I|Λ(|I|)is finite for every interval I ∈ B(T). Fix

I ∈ B(T) such that 1 /∈ I.If I ⊂ Ik,k+1, then

µN(I) =xk,k+1

|Ik,k+1||I| ≤

xk,k+1

|Ik,k+1|Λ(|Ik,k+1|)|I|Λ(|I|) ≤M |I|Λ(|I|).

If I = Ik,l, then

µN(Ik,l) =l−1∑

s=k

µN(Is,s+1) =l−1∑

s=k

xs,s+1 = xk,l ≤M |Ik,l|Λ(|Ik,l|).

Otherwise, denote by Ik,l the largest interval such that Ik,l ⊂ I. We have

µN(I) = µN(Ik,l) + µN(I \ Ik,l)

≤ M |Ik,l|Λ(|Ik,l|) +M |Ik−1,k|Λ(|Ik−1,k|) +M |Il,l+1|Λ(|Il,l+1|)

≤ 3M |I|Λ(|I|).

Thus, µN is a Λ-bounded premeasure. Next, using a Helly-type selection theorem forpremeasures due to Cyphert and Kelingos [13, Theorem 2], we can find a Λ-boundedpremeasure ν and a subsequence µNk

∈ B+Λ such that µNk

k converge Λ-weakly to ν.Furthermore, ν satisfies the following conditions :

34

ν(J) ≤ 3M |J |Λ(|J |) and µ(J) + ν(J) ≤ minC|J |Λ(|J |), ε for every interval J ⊂T \ 1.

Now, if I is an interval containing the point 1, we can represent it as I = I1 ⊔ 1 ⊔ I2,for some (possibly empty) intervals I1 and I2. Then

µ(I) + ν(I) = (µ+ ν)(I1) + (µ+ ν)(I2) + (µ+ ν)(1)

≤ (µ+ ν)(I1) + (µ+ ν)(I2).

Therefore, for every I ∈ B(T) we have µ(J)+ν(J) ≤ 2ε. Since (µ+ν)(T\I) = −µ(I)−ν(I),we have

|µ(J) + ν(J)| ≤ 2ε.

Thus µ is Λ-absolutely continuous.

Lemma 2.11 Let µ ∈ B+Λ be not Λ-absolutely continuous. Then for every C > 0 there is

ε > 0 such that for all M > 0, there exists a simple covering of T by a finite number ofhalf-open intervals Inn, satisfying the relation

∑

n

min µ(In) +M |In|Λ(|In|), C|In|Λ(|In|), ε < 0.

Proof. By Lemma 2.10, for every C > 0 there exists a number ε > 0 such that for allM > 0, the system (2.6) has no solutions for some N ∈ N. In other words, there are noxk,lk,l such that :

l−1∑

s=k

µ(Is,s+1) + xs,s+1 ≤ minµ(Ik,l) +M |Ik,l|Λ(|Ik,l|), C|Ik,l|Λ(|Ik,l|), ε

(2.7)

with xk,l =l−1∑s=k

xs,s+1 and x0,N = 0.

We set Xj = µ(Ij,j+1) + xj,j+1, and

bk,l = minµ(Ik,l+1) +M |Ik,l+1|Λ(|Ik,l+1|), C|Ik,l+1|Λ(|Ik,l+1|), ε

.

Then relations (2.7) are rewritten as

l∑

j=k

Xj ≤ bk,l, 0 ≤ k < l ≤ N − 1.

Therefore, we are in the conditions of Lemma 2.9 with variables Xj. We conclude thatthere is a simple covering of the circle T by a finite number of half-open intervals In suchthat ∑

n

minµ(In) +M |In|Λ(|In|), C|In|Λ(|In|), ε

< 0.

35

In the following lemma we give a normal families type result for the Λ-Carleson sets.

Lemma 2.12 Let Fnn be a sequence of sets on the unit circle, and let each Fn be a finiteunion of closed intervals. We assume that

(i) |Fn| → 0, n→ ∞,

(ii) EntrΛ(Fn) = O(1), n→ ∞.

Then there exists a subsequence Fnkk and a Λ-Carleson set F such that :

for every δ > 0 there is a natural number N with

(a) Fnk⊂ F δ

(b) F ⊂ F δnk.

for all k ≥ N .

Proof. Let Ik,nk be the complementary arcs to Fn such that |I1,n| ≥ |I2,n| ≥ . . .. Weshow first that the sequence |I1,n|n is bounded away from zero. Since the function Λ isnon-increasing, we have

EntrΛ(Fn) =∑

k

|Ik,n|Λ(|Ik,n|) ≥ |T \ Fn|Λ(|I1,n|),

and therefore,EntrΛ(Fn)

|T \ Fn|≥ Λ(|I1,n|).

Now the conditions (i) and (ii) of lemma and the fact that Λ(0+) = +∞ imply that thesequence |I1,n|n is bounded away from zero.

Given a subsequence F(m)k k of Fn, we denote by (I

(m)j,k )j the complementary arcs to

F(m)k . Let us choose a subsequence F

(1)k k such that

I(1)1,k = (a

(1)k , b

(1)k ) → (a1, b1) = J1

as k → +∞, where J1 is a non-empty open arc.If |J1| = 1, then F = T \ J1 is a Λ-Carleson set, and we are done : we can take

Fnkk = F

(1)k k.

Otherwise, if |J1| < 1, then, using the above method we show that

Λ(|I(1)2,k |) ≤

EntrΛ(F(1)k )

|T \ F(1)k | − |I

(1)1,k |

.

Since limk→+∞ |T \ F(1)k | − |I

(1)1,k | = 1 − |J1| > 0, the sequence Λ(|I

(1)2,k |) is bounded,

and hence, the sequence |I(1)2,k | is bounded away from zero. Next we choose a subsequence

36

F(2)k k of F (1)

k k such that the arcs I(2)2,k = (a2k, b2k) tend to (a(2), b(2)) = J2, where J2 is a

non-empty open arc. Repeating this process we can have two possibilities. First, supposethat after a finite number of steps we have |J1| + . . . + |Jm| = 1, and then we can takeFnk

k = F(m)k k,

I(m)j,k → Jj, 1 ≤ j ≤ m,

as k → +∞, and F = T \m∪j=1Jj is Λ-Carleson.

Now, if the number of steps is infinite, then using the estimate

Λ(|Jl|) ≤supn

EntrΛ(Fn)

1−∑l−1

k=1 |Jk|,

and the fact |Jm| → 0 as m→ ∞, we conclude that

∞∑

j=1

|Jj| = 1.

We can set Fnkk = F

(m)m m, F = T \

⋃j≥1 Jj.

In all three situations the properties (a) and (b) follow automatically.

Proof of Theorem 2.8

First we suppose that µ is Λ-absolutely continuous, and prove that µs = 0. Choosea sequence µn of Λ-bounded premeasures satisfying the properties (1) and (2) of Defini-tion 2.7. Let F be a Λ-Carleson set and let (In)n be the sequence of the complementaryarcs to F . Denote by (µ+ µn)s the Λ-singular part of µ+ µn. Then

−(µ+ µn)s(F ) =∑

k

(µ+ µn)(Ik)

=∑

k≤N

(µ+ µn)(Ik) +∑

k>N

(µ+ µn)(Ik)

≤∑

k≤N

(µ+ µn)(Ik) + C∑

k>N

|Ik|Λ(|Ik|)

Using the property (2) of Definition 2.7 we obtain that

− lim infn→∞

(µ+ µn)s(F ) ≤ C∑

k>N

|Ik|Λ(|Ik|).

Since F ∈ CΛ, we have∑

k>N |Ik|Λ(|Ik|) → 0 as N → +∞, and hence lim infn→∞(µ +µn)s(F ) ≥ 0. Since (µ + µn) ∈ B+

Λ , by Proposition 2.5 its Λ-singular part is non-positive.Thus limn→∞(µ+ µn)s(F ) = 0 for all F ∈ CΛ, which proves that µs = 0.

37

Now, let us suppose that µ is not Λ-absolutely continuous. We apply Lemma 2.11 withC = 4‖µ‖+Λ and find ε > 0 such that for all M > 0, there is a simple covering of circle T

by a half-open intervals I1, I2, . . . IN such that

∑

n

minµ(In) +M |In|Λ(|In|), 4‖µ‖

+Λ |In|Λ(|In|), ε

< 0. (2.8)

Let us fix a number ρ > 0 satisfying the inequality ρΛ(ρ) ≤ ε/4‖µ‖+Λ . We divide theintervals I1, I2, . . . IN into two groups. The first group I

(1)n n consists of intervals In

such that

minµ(In) +M |In|Λ(|In|), 4‖µ‖+Λ |In|Λ(|In|), ε = µ(In) +M |In|Λ(|In|), (2.9)

and the second one is I(2)n n = Inn \ I

(1)n n.

Using these definitions and the fact that Λ is non-increasing, we rewrite inequality (2.8)as

∑

n

µ(I(1)n ) +M∑

n

|I(1)n |Λ(|I(1)n |)

< −4‖µ‖+Λ∑

n : |I(2)n |<ρ

|I(2)n |Λ(|I(2)n |)− ε Cardn : |I(2)n | ≥ ρ. (2.10)

Next we establish three properties of these families of intervals. From now on we assumethat M > 4‖µ‖+Λ .

(1) We have I(2)n : |I

(2)n | ≥ ρ 6= ∅. Otherwise, by (2.10), we would have

0 = µ(T) =∑

n

µ(I(1)n ) +∑

n

µ(I(2)n )

≤ −M∑

n

|I(1)n |Λ(|I(1)n |)− 4‖µ‖+Λ∑

n

|I(2)n |Λ(|I(2)n |) + ‖µ‖+Λ∑

n

|I(2)n |Λ(|I(2)n |)

≤ −M∑

n

|I(1)n |Λ(|I(1)n |)− 3‖µ‖+Λ∑

n

|I(2)n |Λ(|I(2)n |) < 0.

(2) We have∑

n |I(2)n |Λ(|I

(2)n |) ≤ 2Λ(ρ). To prove this relation, we notice first that for

every simple covering Jnn of T, we have

0 = µ(T) =∑

n

µ(Jn) =∑

n

µ(Jn)+ −

∑

n

µ(Jn)−,

and hence,∑

n

|µ(Jn)| =∑

n

µ(Jn)+ +

∑

n

µ(Jn)− = 2

∑

n

µ(Jn)+ ≤ 2‖µ‖+Λ

∑

n

|Jn|Λ(|Jn|).

38

Applying this to our simple covering, we get∑

n

|µ(I(1)n )|+∑

n

|µ(I(2)n )| ≤ 2‖µ‖+Λ∑

n

[|I(1)n |Λ(|I(1)n |) + |I(2)n |Λ(|I(2)n |)

],

and hence,

−∑

n

µ(I(1)n ) ≤ 2‖µ‖+Λ∑

n

[|I(1)n |Λ(|I(1)n |) + |I(2)n |Λ(|I(2)n |)

].

Now, using (2.10) we obtain that

M∑

n

|I(1)n |Λ(|I(1)n |)+ 4‖µ‖+Λ∑

|I(2)n |<ρ

|I(2)n |Λ(|I(2)n |) ≤ 2‖µ‖+Λ∑

n

[|I(1)n |Λ(|I(1)n |)+ |I(2)n |Λ(|I(2)n |)

],

and hence,

(M−2‖µ‖+Λ

)∑

n

|I(1)n |Λ(|I(1)n |) ≤ 2‖µ‖+Λ

[ ∑

|I(2)n |≥ρ

|I(2)n |Λ(|I(2)n |)−∑

|I(2)n |<ρ

|I(2)n |Λ(|I(2)n |)

]. (2.11)

As a consequence, we have∑

|I(2)n |<ρ

|I(2)n |Λ(|I(2)n |) ≤∑

|I(2)n |≥ρ

|I(2)n |Λ(|I(2)n |),

and, finally,∑

n

|I(2)n |Λ(|I(2)n |) ≤ 2∑

|I(2)n |≥ρ

|I(2)n |Λ(|I(2)n |) ≤ 2∑

n

|I(2)n |Λ(ρ) ≤ 2Λ(ρ).

(3) We have∑

n

|I(1)n |Λ(|I(1)n |) ≤2‖µ‖+Λ

M − 2‖µ‖+Λ· Λ(ρ).

This property follows immediately from (2.11).

We set FM =⋃n I

(1)n . Inequality (2.10) and the properties (1)–(3) show that

(i) EntrΛ(FM) = O(1), M → ∞,

(ii) |FM |Λ(|FM |) ≤2‖µ‖+Λ

M−2‖µ‖+Λ· Λ(ρ),

(iii) µ(FM) ≤ −4‖µ‖+Λ

[∑n

|I(1)n |Λ(|I

(1)n |) +

∑n : |I

(2)n |<ρ

|I(2)n |Λ(|I

(2)n |)

]− ε.

By Lemma 2.12 there exists a subsequence Mn → +∞ such that F ∗n := FMn (composed of

a finite number of closed arcs) converge to a Λ-Carleson set F . More precisely, F ⊂ F ∗nδ

and F ∗n ⊂ F δ for every fixed δ > 0 and for sufficiently large n. Furthermore, (iii) yields

µ(F ∗n) ≤ −4‖µ‖+Λ

[∑

k

|Rk,n|Λ(|Rk,n|) +∑

k:|Lk,n|<ρ

|Lk,n|Λ(|Lk,n|)

]− ε, (2.12)

39

where F ∗n =

⊔k Rk,n and T \ F ∗

n =⊔k Lk,n.

It remains to show thatµs(F ) < 0.

Otherwise, if µs(F ) = 0, then by Proposition 2.6 we have

limδ→0

µ(F δ) = 0.

Modifying a bit the set F ∗n , if necessary, we obtain limδ→0 µ(F

∗n ∩ F δ) = 0. Now we can

choose a sequence δn > 0 rapidly converging to 0 and a sequence kn rapidly convergingto ∞ such that the sets Fn defined by

Fn = F ∗kn \ F δn+1 ⊂ F δn \ F δn+1 ,

and consisting of a finite number of intervals Ik,nk satisfy the inequalities

µ(Fn) ≤ −4‖µ‖+Λ

[∑

k

|Ik,n|Λ(|Ik,n|) +∑

k

|Jn,k|Λ(|Jn,k|)

]− ε/2, (2.13)

where⊔k Jn,k = (F δn \ F δn+1) \ Fn =: Gn.

We denote by In, Jn, and Kn the systems of intervals that form Fn, Gn, and F δn ,respectively. Furthermore, we denote by I0 be the system of intervals complementary to

F δ1 , and we put Sn = (n∪k=1

Ik) ∪ (n∪k=1

Jn) ∪ Kn+1. Summing up the estimates on µ(Fn) in

(2.13) we obtain

∑

I∈I0

|µ(I)|+∑

I∈Sn

|µ(I)| ≥n∑

i=1

|µ(Fi)|

≥ 4‖µ‖+Λ

n∑

i=1

[∑

k

|Ii,k|Λ(|Ii,k|) +∑

k

|Ji,k|Λ(|Ji,k|)

]+ nε/2

= 4‖µ‖+Λ∑

I∈Sn

|I|Λ(|I|)− 4‖µ‖+Λ∑

I∈Kn+1

|I|Λ(|I|) + nε/2

= 4‖µ‖+Λ

[∑

I∈Sn∪I0

|I|Λ(|I|)−∑

I∈Kn+1

|I|Λ(|I|)

]

− 4‖µ‖+Λ∑

I∈I0

|I|Λ(|I|) + nε/2. (2.14)

Notice that∑

I∈Kn+1

|I|Λ(|I|) ≤∑

|Jk|<2δn+1

|Jk|Λ(|Jk|) + 2δn+1Λ(δn+1) · Cardk : |Jk| ≥ 2δn+1,

40

where Jkk, |J1| ≥ |J2| ≥ . . . are the complementary arcs to the Λ-Carleson set F . Sincelimt→0 tΛ(t) = 0, we obtain that

limn→+∞

∑

I∈Kn+1

|I|Λ(|I|) = 0.

Thus for sufficiently large n, (2.14) gives us the following relation∑

I∈Sn∪I0

|µ(I)| ≥ 4‖µ‖+Λ∑

I∈Sn∪I0

|I|Λ(|I|)

where Sn ∪ I0 is a simple covering of the unit circle. However, since µ ∈ B+Λ , we have

∑

I∈Sn∪I0

|µ(I)| = 2∑

I∈Sn∪I0

max(µ(I), 0) ≤ 2‖µ‖+Λ∑

I∈Sn∪I0

|I|Λ(|I|).

This contradiction completes the proof of the theorem.

2.2 Harmonic functions of restricted growth

Every bounded harmonic function can be represented via the Poisson integral of itsboundary values. In the following theorem we show that a large class of real-valued harmo-nic functions in the unit disk D can be represented as the Poisson integrals of Λ-boundedpremeasures. Before formulating the main result of this section, let us introduce somenotations.

Definition 2.13 Let f be a function in C1(T) and let µ ∈ B+Λ . We define the integral of

the function f with respect to µ by the formula

∫

T

f dµ =

∫ 2π

0

f(eit) dfµ(t).

In particular, we have

∫ 2π

0

1− |z|2

|eiθ − z|2dµ(θ) = −

∫ 2π

0

( ∂

∂θ

1− |z|2

|eiθ − z|2

)fµ(θ) dθ.

Given a Λ-bounded premeasure µ we denote by P [µ] its Poisson integral :

P [µ](z) =

∫ 2π

0

1− |z|2

|eiθ − z|2dµ(θ).

Proposition 2.14 Let µ ∈ B+Λ . The Poisson integral P [µ] satisfies the estimate

P [µ](z) ≤ 10‖µ‖+ΛΛ(1− |z|), z ∈ D.

41

Proof. It suffices to verify the estimate on the interval (0, 1). Let 0 < r < 1. Then

P [µ](r) =

∫ 2π

0

1− r2

|eiθ − r|2dµ(θ)

= −

∫ 2π

0

[ ∂∂θ

( 1− r2

|eiθ − r|2)]fµ(θ) dθ

=

∫ 2π

0

2r(1− r2) sin θ

(1− 2r cos θ + r2)2µ(Iθ) dθ

=

∫ π

0

2r(1− r2) sin θ

(1− 2r cos θ + r2)2µ(Iθ) dθ −

∫ 0

π

−2r(1− r2) sin θ

(1− 2r cos θ + r2)2µ(I2π−θ) dθ

=

∫ π

0

2r(1− r2) sin θ

(1− 2r cos θ + r2)2

[µ(Iθ) + µ([−θ, 0))

]dθ

=

∫ π

0

2r(1− r2) sin θ

(1− 2r cos θ + r2)2µ([−θ, θ)) dθ.

Integrating by parts and using the fact that Λ is decreasing and tΛ(t) is increasing weget

P [µ](r) ≤ ‖µ‖+ΛΛ(1−r)

[(1−r)

∫ 1−r2

0

2r(1− r2) sin θ

(1− 2r cos θ + r2)2dθ−

∫ π

1−r2

2θ[ ∂∂θ

( 1− r2

|eiθ − r|2)]dθ

]

≤ ‖µ‖+ΛΛ(1− r)

[2(1− r)3

∫ 1−r2

0

dθ

(1− r)4+

(1− r)(1− r2)

(1− r)2+ 2

∫ π

0

1− r2

|eiθ − r|2dθ

]

≤ 10‖µ‖+ΛΛ(1− r).

The following theorem is stated by Korenblum in [36, Theorem 1, p. 543] without proof,in a more general situation.

Theorem 2.15 Let h be a real-valued harmonic function on the unit disk such that h(0) =0 and

h(z) = O(Λ(1− |z|)), |z| → 1, z ∈ D.



µ(I) = limr→1−


∫

I

h(rξ) |dξ| <∞.

42

2. µ is a Λ-bounded premeasure.


h(z) =

∫ 2π

0

1− |z|2

|eiθ − z|2dµ(θ), z ∈ D.

Proof. Let

h(reiθ) =+∞∑

n=−∞

anr|n|einθ.

Since a0 = h(0) = 0, we have∫ 2π

0

h+(reiθ) dθ =

∫ 2π

0

h−(reiθ) dθ =1

2

∫ 2π

0

|h(reiθ)| dθ.

Furthermore,

|an| =∣∣∣r

−|n|

2π

∫ 2π

0

h(reiθ)e−inθ dθ∣∣∣

≤r−|n|

2π

∫ 2π

0

|h(reiθ)| dθ =r−|n|

π

∫ 2π

0

h+(reiθ) dθ

≤ Cr−|n|Λ(1− r)

≤ C1Λ( 1

|n|

),

1

|n|= 1− r, n ∈ Z \ −1, 0, 1. (2.15)

Let I = eiθ : α ≤ θ ≤ β be an arc of T, τ = β − α. For θ ∈ [α, β] we define

t(θ) = minθ − α, β − θ, η(θ) =1

τ(β − θ)(θ − α).

Then1

2t(θ) ≤ η(θ) ≤ t(θ), |η′(θ)| ≤ 1, η′′(θ) =

−2

τ, θ ∈ [α, β].

Given p > 2 we introduce the function q(θ) = 1−η(θ)p satisfying the following properties :

|q′(θ)| ≤ pη(θ)p−1, |q′′(θ)| ≤ p2η(θ)p−2, θ ∈ (α, β).

Integrating by parts we obtain for |n| ≥ 1 and τ < 1 that

∣∣∣∫ β

α

(1− q(θ)|n|)einθ dθ∣∣∣ = 1

|n|

∣∣∣∫ β

α

|n|q(θ)|n|−1q′(θ)einθ dθ∣∣∣

≤|n| − 1

|n|

∫ β

α

q(θ)|n|−2|q′(θ)|2 dθ +1

|n|

∫ β

α

q(θ)|n|−1|q′′(θ)| dθ

≤ 2p2∫ τ/2

0

(1−

[ t2

]p)|n|−2

t2p−2 dt+2p2

|n|

∫ τ/2

0

(1−

[ t2

]p)|n|−1

tp−2 dt

≤ Cp

[∫ τ/4

0

(1− tp

)|n|−2t2p−2 dt+

1

|n|

∫ τ/4

0

(1− tp

)|n|−1tp−2 dt

],

43

and, hence,

∣∣∣∫ β

α

(1− q(θ)|n|)einθ dθ∣∣∣ ≤ C1,pτ max

0≤t≤1

(1− tp

)|n|−2t2p−2 +

1

|n|

(1− tp

)|n|−1tp−2

≤ C2,pτ |n|−2(1− 1

p).

On the other hand, we have

1

2π

∫

I

h(rξ) |dξ| =1

2π

∫ β

α

h(rq(θ)eiθ) dθ +1

2π

∫ β

α

[h(reiθ)− h(rq(θ)eiθ)

]dθ.

By (2.15), we obtain

∣∣∣ 12π

∫ β

α


]dθ∣∣∣ ≤

1

2π

∑

n∈Z

|an|∣∣∣∫ β

α

r|n|(1− q(θ)|n|)einθ dθ∣∣∣

≤ C3,pτ∑

n∈Z

|an|(|n|+ 1)−2(1− 1p)

≤ C4,pτ∑

n∈Z

Λ( 1

max(|n|, 1)

)(|n|+ 1)−2(1− 1

p).

Therefore, if t 7→ tαΛ(t) increase, and

α +2

p< 1, (2.16)

then ∣∣∣ 12π

∫ β

α


]dθ∣∣∣ ≤ C5,pτ.

Since Λ(xp) ≤ CpΛ(x), we obtain

∣∣∣ 12π

∫ β

α

h(rq(θ)eiθ) dθ∣∣∣ ≤ C

∫ β

α

Λ(1− q(θ)) dθ

≤ C

∫ β

α

Λ(t(θ)

2

)dθ

≤ C1

∫ τ/4

0

Λ(t) dt

= C1

∫ τ/4

0

t−αtαΛ(t) dt

≤ C2ταΛ(τ)

∫ τ/4

0

t−α dt

= C3τΛ(τ).

44

Hence,µr(I) ≤ C|I|Λ(|I|)

for some C independent of I.Given r ∈ (0, 1), we define hr(z) = h(rz). The hr is the Poisson integral of dµr =

hr(eiθ) dθ :

hr(z) =

∫

T

1− |z|2

|eiθ − z|2dµr(θ)

The set µr : r ∈ (0, 1) is a uniformly Λ-bounded family of premeasures. Using a Helly-type theorem [31, Theorem 1, p. 204], we can find a sequence of premeasures µrn ∈ B+

Λ

converging weakly to a Λ-bounded premeasure µ as n→ ∞, limn→∞ rn = 1. Then

µ(I) ≤ C|I|Λ(|I|)

for every arc I, and

hrn(z) = −

∫ 2π

0

∂

∂θ

( 1− |z|2

|eiθ − z|2

)fµn(θ) dθ.

Passing to the limit we conclude that

h(z) =

∫

T

1− |z|2

|eiθ − z|2dµ(θ).

2.3 Cyclic vectors

Given a Λ-bounded premeasure µ, we consider the corresponding analytic fuction

fµ(z) = exp

∫ 2π

0

eiθ + z

eiθ − zdµ(θ). (2.17)

If µ is a positive singular measure on the circle T, we denote by Sµ the associated singularinner function. Notice that in this case µ = µ(T)m − µ is a premeasure, and we haveSµ = fµ/Sµ(0) ; m is (normalized) Lebesgue measure.

Let f be a zero-free function in A−∞Λ such that f(0) = 1. According to Theorem 2.15,

there is a premeasure µf ∈ B+Λ such that

f(z) = exp

∫ 2π

0

eiθ + z

eiθ − zdµf (θ).

The following result follows immediately from Theorem 2.8.

Theorem 2.16 Let f ∈ A−∞Λ be a zero-free function such that f(0) = 1. If (µf )s ≡ 0,

then f is cyclic in A−∞Λ .

45

Proof. Suppose that (µf )s ≡ 0. By theorem 2.8, µf is Λ-absolutely continuous. Letµnn≥1 be a sequence of Λ-bounded premeasures from Definition 2.7. We set

gn(z) = exp

∫ 2π

0

eiθ + z

eiθ − zdµn(θ), z ∈ D.

By Proposition 2.14, gn ∈ A−∞Λ , and

f(z)gn(z) = exp

∫ 2π

0

eiθ + z

eiθ − zd(µf + µn)(θ)

= exp[−

∫ 2π

0

∂

∂θ

(eiθ + z

eiθ − z

)[fµn(θ)−

fµ(θ)] dθ

]

= exp[−

∫ 2π

0

∂

∂θ

(eiθ + z

eiθ − z

)[µ(Iθ) + µn(Iθ)

]dθ].

Again by Definition 2.7, we obtain that f(z)gn(z) → 1 uniformly on compact subsets ofunit disk D. This yields that fgn → 1 in A−∞

Λ as n→ ∞.

From now on, we deal with the statements converse to Theorem 2.16. We’ll establishtwo results valid for different growth ranges of the majorant Λ. More precisely, we considerthe following growth and regularity assumptions :

for every c > 0, the function x 7→ exp[cΛ(1/x)

]is concave for large x, (C1)

limt→0

Λ(t)

log(1/t)= ∞. (C2)




(log(1/x))p, p > 1.

Thus, we consider majorants which grow less rapidly than the Korenblum majorant (Λ(x) =log(1/x)) in Case 1 or more rapidly than the Korenblum majorant in Case 2.

2.3.1 Weights Λ satisfying condition (C1)

We start with the following observation :

Λ(t) = o(log 1/t), t→ 0.

Next we pass to some notations and auxiliary lemmas. Given a function f in L1(T), wedenote by P [f ] its Poisson transform,

P [f ](z) =1

2π

∫ 2π

0

1− |z|2

|eiθ − z|f(eiθ) dθ, z ∈ D.

46

Denote by A(D) the disk-algebra, i.e., the algebra of functions continuous on the closed unitdisk and holomorphic in D. A positive continuous increasing function ω on [0,∞) is said tobe a modulus of continuity if ω(0) = 0, t 7→ ω(t)/t decreases near 0, and limt→0 ω(t)/t = ∞.Given a modulus of continuity ω, we consider the Lipschitz space Lipω(T) defined by

Lipω(T) = f ∈ C(T) : |f(ξ)− f(ζ)| ≤ C(f)ω(|ξ − ζ|).

Since the function t 7→ exp[2Λ(1/t)] is concave for large t, and Λ(t) = o(log(1/t)),t → 0, we can apply a result of Kellay [26, Lemma 3.1], to get a non-negative summablefunction ΩΛ on [0, 1] such that

e2Λ(1

n+1) − e2Λ(

1n) ≍

∫ 1

1− 1n

ΩΛ(t)dt, n ≥ 1.

Next we consider the Hilbert space L2ΩΛ

(T) of the functions f ∈ L2(T) such that

‖f‖2ΩΛ= |P [f ](0)|2 +

∫

D

P [|f |2](z)− |P [f ](z)|2

1− |z|2ΩΛ(|z|) dA(z) <∞,

where dA denote the normalized area measure. We need the following lemma.

Lemma 2.17 Under our conditions on Λ and ΩΛ, we have

1. ‖f‖2ΩΛ≍

∑n∈Z |f(n)|

2e2Λ(1/n), f ∈ L2ΩΛ

(T),

2. the functions exp(−cΛ(t)) are moduli of continuity for c > 0,

3. for some positive a, the function ρ(t) = exp(− 32aΛ(t)) satisfies the property

Lipρ(T) ⊂ L2ΩΛ

(T).

For the first statement see [11, Lemma 6.1] (where it is attributed to Aleman [1]) ; thesecond statement is [11, Lemma 8.4] ; the third statement follows from [11, Lemmas 6.2and 6.3].

Recall that

A−1Λ = f ∈ Hol(D) : |f(z)| ≤ C(f) exp(Λ(1− |z|)).

Lemma 2.18 Under our conditions on Λ, there exists a positive number c such that

P+Lipe−cΛ(T) ⊂ (A−1Λ )∗

via the Cauchy duality

〈f, g〉 =∑

n≥0

ang(n),

where f(z) =∑

n≥0 anzn ∈ A−1

Λ , g ∈ Lipe−cΛ(T), and P+ is the orthogonal projector fromL2(T) onto H2(D).

47

Proof. Denote

L2Λ(D) =

f ∈ Hol(D) :

∫

D

|f(z)|2|Λ′(1− |z|)|e−2Λ(1−|z|) dA(z) < +∞,

andB2Λ =

f(z) =

∑

n≥0

anzn : |a0|

2 +∑

n>0

|an|2e−2Λ(1/n) <∞

.

Let us prove thatL2Λ(D) = B2

Λ. (2.18)

To verify this equality, it suffices sufficient to check that

e−2Λ(1/n) ≍

∫ 1

0

r2n+1|Λ′(1− r)|e−2Λ(1−r) dr.

In fact,∫ 1

1−1/n

r2n+1|Λ′

(1− r)|e−2Λ(1−r) dr ≍

∫ 1

1−1/n

|Λ′

(1− r)|e−2Λ(1−r) dr ≍ e−2Λ( 1n), n ≥ 1.

On the other hand,∫ 1−1/n

0

r2n+1|Λ′(1− r)|e−2Λ(1−r) dr = −

∫ 1−1/n

0

r2n+1 de−2Λ(1−r)

≍ −e−2Λ(1/n) + (2n+ 1)

∫ 1−1/n

0

r2ne−2Λ(1−r)dr

≍ n

n∑

k=1

e−2n/ke−2Λ(1/k) 1

k2.

Since the function exp[2Λ(1/x)

]is concave, we have e2Λ(1/k) ≥ k

ne2Λ(1/n), and hence,

e−2Λ(1/k) ≤n

ke−2Λ(1/n).

Therefore,

∫ 1−1/n

0

r2n+1|Λ′(1− r)|e−2Λ(1−r) dr ≤ Cn2e−2Λ(1/n)

n∑

k=1

e−2n/k 1

k3≍ e−2Λ(1/n),

and (2.18) follows.Since A−1

Λ ⊂ L2Λ(D), we have (B2

Λ)∗ ⊂ (A−1

Λ )∗. By Lemma 2.17, we have P+Lipρ(T) ⊂(B2

Λ)∗. Thus,

P+Lipρ(T) ⊂ (A−1Λ )∗.

48

Lemma 2.19 Let f ∈ A−nΛ for some n > 0. The function f is cyclic in A−∞

Λ if and onlyif there exists m > n such that f is cyclic in A−m

Λ .

Proof. Notice that the space A−∞Λ is endowed with the inductive limit topology indu-

ced by the spaces A−NΛ . A sequence fnn ∈ A−∞

Λ converges to g ∈ A−∞Λ if and only if

there exists N > 0 such that all fn and g belong to A−NΛ , and limn→+∞ ‖fn − g‖A−N

Λ= 0.

The statement of the lemma follows.



Proof. Suppose that the Λ-singular part µs of µ is non-trivial. There exists a Λ-Carlesonset F ⊂ T such that −∞ < µs(F ) < 0. We set ν = −µs

∣∣F . By a theorem of Shirokov [46,Theorem 9, pp.137,139], there exists an outer function ϕ such that

ϕ ∈ Lipρ(T) ∩ H∞(D), ϕSν ∈ Lipρ(T) ∩ H∞(D),

and the zero set of the function ϕ coincides with F . Next, for ξ, θ ∈ [0, 2π] we have

|ϕSν(eiξ)− ϕSν(e

iθ)| = |ϕ(eiξ)Sν(eiθ)− ϕ(eiθ)Sν(e

iξ)|

≤ |(ϕ(eiξ)− ϕ(eiθ)

)Sν(e

iθ)|+ |(ϕ(eiθ)− ϕ(eiξ)

)Sν(e

iξ)|

+ |(ϕSν)(eiθ)− (ϕSν)(e

iξ)|,

and hence,ϕSν ∈ Lipρ(T).

Set g = P+

(zϕSν

). Since ϕSν ∈ Lipρ(T), we have g ∈ (A−1

Λ )∗. Consider the followinglinear functional on A−1

Λ :

Lg(f) = 〈f, g〉 =∑

n≥0

ang(n), f(z) =∑

n≥0

anzn ∈ A−1

Λ .

Suppose that Lg = 0. Then, for every n ≥ 0 we have

0 = Lg(zn)

=

∫ 2π

0

einθg(eiθ)dθ

2π

=

∫ 2π

0

ei(n+1)θ ϕ(eiθ)

Sν(eiθ)

dθ

2π.

We conclude that ϕ/Sν ∈ H∞(D), which is impossible. Thus, Lg 6= 0.

49

On the other hand we have, for every n ≥ 0,

Lg(znSν) =

∫ 2π

0

einθSν(eiθ)g(eiθ)

dθ

2π

=

∫ 2π

0

einθSν(eiθ)g(eiθ)

dθ

2π

=

∫ 2π

0

ei(n+1)θϕ(eiθ)dθ

2π= 0.

Thus, g ⊥ [fµ]A−1Λ

which implies that the function fµ is not cyclic in A−1Λ . By Lemma 2.19,

fµ is not cyclic in A−∞Λ .

2.3.2 Weights Λ satisfying condition (C2)

We start with an elementary consequence of the Cauchy formula.

Lemma 2.21 Let f(z) =∑n≥0

anzn be an analytic function in D. If f ∈ A−∞

Λ , then there

exists C > 0 such that

|an| = O(exp[CΛ(1

n)]) as n→ +∞.



Proof. We define

A∞Λ =

⋂

c<∞

g ∈ Hol(D) ∩ C∞(D) : |f(n)| = O(exp[−cΛ(

1

n)])

,

and, using Lemma 2.21, we obtain that A∞Λ ⊂ (A−∞

Λ )∗ via the Cauchy duality

〈f, g〉 =∑

n≥0

f(n)g(n) = limr→1

∫ 2π

0

f(rξ)g(ξ)dξ, f ∈ A−∞Λ , g ∈ A∞

Λ .

Suppose that the Λ-singular part µs of µ is nonzero. Then there exists a Λ-Carleson setF ⊂ T such that −∞ < µs(F ) < 0. We set σ = µs

∣∣F . By a theorem of Bourhim, El-Fallah,and Kellay [11, Theorem 5.3] (extending a result of Taylor and Williams), there exist anouter function ϕ ∈ A∞

Λ such that the zero set of ϕ and of all its derivatives coincides exactlywith the set F , a function Λ such that

Λ(t) = o(Λ(t)), t→ 0, (2.19)

50

and a positive constant B such that

|ϕ(n)(z)| ≤ n!BneΛ∗(n), n ≥ 0, z ∈ D, (2.20)

where Λ∗(n) = supx>0

nx− Λ(e−x/2)

.

We setΨ = ϕSσ.

For some positive D we have

|S(n)σ (z)| ≤

Dnn!

dist(z, F )2n, z ∈ D, n ≥ 0. (2.21)

By the Taylor formula, for every n, k ≥ 0, we have

|ϕ(n)(z)| ≤1

k!dist(z, F )kmax

w∈D|ϕ(n+k)(w)|, z ∈ D. (2.22)

Next, integrating by parts, for every n 6= 0, k ≥ 0 we obtain

|Ψ(n)| =∣∣(ϕSσ)(n)

∣∣ = 1

2π

∣∣∣∫ 2π

0

(ϕSσ)(k)(eit)

nke−int dt

∣∣∣.

Applying the Leibniz formula and estimates (2.20)–(2.22), we obtain for n ≥ 1 that

∣∣Ψ(n)∣∣ ≤ inf

k≥0

1

nkmaxt∈[0,2π]

∣∣(ϕSσ)(k)(eit)∣∣

≤ infk≥0

1

nk

k∑

s=0

Csk maxt∈[0,2π]

|S(s)σ (eit)| max

t∈[0,2π]|ϕ(k−s)(eit)|

≤ infk≥0

1

nk

k∑

s=0

CskD

ss!1

(2s)!(k + s)!Bk+seΛ

∗(k+s)

≤ infk≥0

eΛ

∗(2k)(B2D

n

)k k∑

s=0

(k + s)!k!

(2s)!(k − s)!

≤ infk≥0

k!eΛ

∗(2k)(4B2D

n

)k

≤ infk≥0

k!(4B2D

n

)ksup0<t<1

e−Λ(t1/4)t−k

.

By property (2.19), for every C > 0 there exists a positive number K such that

e−Λ(t1/4) ≤ Ke−Λ(Ct), t ∈ (0, 1).

51

We take C = 18B2D

, and obtain for n 6= 0 that

∣∣Ψ(n)∣∣ ≤ K inf

k≥0

(4B2D

n

)kk! sup

0<t<1

e−Λ(Ct)

tk

≤ K1 infk≥0

(2n)−kk! sup

0<t<1

e−Λ(t)

tk

.

Finally, using [28, Lemma 6.5] (see also [11, Lemma 8.3]), we get

|Ψ(n)| = O(e−Λ(1/n)

), |n| → ∞.

Thus, the function g = P+

(zϕSσ

)belongs to (A−1

Λ )∗. Now we obtain that fµ is notcyclic using the same argument as that at the end of Case 1. This concludes the proof ofthe theorem.

Theorems 2.20 and 2.22 together give a positive answer to a conjecture by Deninger[14, Conjecture 42].

We complete this section by two examples that show how the cyclicity property of afixed function changes in a scale of A−∞

Λ spaces.

Example 2.23 Let Λα(x) = (log(1/x))α, 0 < α < 1, and let 0 < α0 < 1. There exists asingular inner function Sµ such that

Sµ is cyclic in A−∞Λα

⇐⇒ α > α0.

Construction. We start by defining a Cantor type set and the corresponding canonicalmeasure. Let mkk≥1 be a sequence of natural numbers. Set Mk =

∑1≤s≤kms, and assume

thatMk ≍ mk, k → ∞. (2.23)

Consider the following iterative procedure. Set I0 = [0, 1]. On the step n ≥ 1 the set In−1

consist of several intervals I. We divide each I into 2mn+1 equal subintervals and replace itby the union of every second interval in this division. The union of all such groups is In.Correspondingly, In consists of 2Mn intervals ; each of them is of length 2−n−Mn . Next, weconsider the probabilistic measure µn equidistributed on In. Finally, we set E = ∩n≥1In,and define by µ the weak limit of the measures µn.

Now we estimate the Λα-entropy of E :

EntrΛα(In) ≍∑

1≤k≤n

2Mk · 2−k−Mk · Λα(2−k−Mk) ≍

∑

1≤k≤n

2−k ·mαk , n→ ∞.

Thus, if ∑

n≥1

2−n ·mα0n <∞, (2.24)

then EntrΛα0(E) <∞. By Theorem 2.20, Sµ is not cyclic in A−∞

Λαfor α ≤ α0.

52

Next we estimate the modulus of continuity of the measure µ,

ωµ(t) = sup|I|=t

µ(I).

Assume thatAj+1 = 2−(j+1)−Mj+1 ≤ |I| < Aj = 2−j−Mj ,

and that I intersects with one of the intervals Ij that constitute Ij. Then

µ(I) ≤ 4|I|

Ajµ(Ij) = 4|I|2j+Mj2−Mj = 4|I|2j.

Thus, if2j ≤ C(log(1/Aj))

α ≍ mαj , j ≥ 1, α0 < α < 1, (2.25)

thenωµ(t) ≤ Ct(log(1/t))α.

By [3, Corollary B], we have µ(F ) = 0 for any Λα-Carleson set F , α0 < α < 1. Againby Theorem 2.20, Sµ is cyclic in A−∞

Λαfor α > α0. It remains to fix mkk≥1 satisfying

(2.23)–(2.25). The choice mk = 2k/α0k−2/α0 works.

Of course, instead of Theorem 2.20 we could use here [11, Theorem 7.1].

Example 2.24 Let Λα(x) = (log(1/x))α, 0 < α < 1, and let 0 < α0 < 1. There exists apremeasure µ such that µs is infinite,

fµ is cyclic in A−∞Λα

⇐⇒ α > α0,

where fµ is defined by (2.17).

It looks like the subspaces [fµ]A−∞Λα

, α ≤ α0, contain no nonzero Nevanlinna class func-tions. For a detailed discussion on Nevanlinna class generated invariant subspaces in theBergman space (and in the Korenblum space) see [23].

For α ≤ α0, instead of Theorem 2.20 we could once again use here [11, Theorem 7.1].

Construction. We use the measure µ constructed in Example 2.23.Choose a decreasing sequence uk of positive numbers such that

∑

k≥1

uk = 1,∑

k≥1

vk = +∞,

where vk = uk log log(1/uk) > 0, k ≥ 1.Given a Borel set B ⊂ B0 = [0, 1], denote

Bk = ukt+k−1∑

j=1

uj : t ∈ B ⊂ [0, 1],

53

and define measures νk supported by B0k by

νk(Bk) =vkukm(Bk)− vkµ(B),

where m(Bk) is Lebesgue measure of Bk.We set

ν =∑

k≥1

νk.

Then ν(B0k) = νk(B

0k) = 0, k ≥ 1, and ν is a premeasure.

Sincevk ≤ C(α)ukΛα(uk), 0 < α < 1,

ν is a Λα-bounded premeasure for α ∈ (0, 1).Furthermore, as above, by Theorem 2.20, fν is not cyclic in A−∞

Λαfor α ≤ α0.

Next, we estimateων(t) = sup

|I|=t

|ν(I)|.

As in Example 2.23, if j, k ≥ 1 and

ukAj+1 ≤ |I| < ukAj,

then|ν(I)|

|I|≤ C · 2j ·

vkuk. (2.26)

Now we verify thatων(t) ≤ Ct(log(1/t))α, α0 < α < 1. (2.27)

Fix α ∈ (α0, 1), and use that

(log

1

Aj

)α≥ C · 2(1+ε)j, j ≥ 1,

for some C, ε > 0. By (2.26), it remains to check that

2j log log1

uk≤ C

(2(1+ε)j +

(log

1

uk

)α).

Indeed, if

log log1

uk> 2εj,

then

C(log

1

uk

)α> 2j log log

1

uk.

Finally, we fix α ∈ (α0, 1) and a Λα-Carleson set F . We have

T \ F = ⊔sL∗s

54

for some intervals L∗s. By [3, Theorem B], there exist disjoint intervals Ln,s such that

F ⊂ ⊔sLn,s,∑

s

|Ln,s|Λα(|Ln,s|) <1

n, n ≥ 1.

Then by (2.27), ∑

s

|ν(Ln,s)| <c

n.

SetT \ ⊔sLn,s = ⊔sL

∗n,s.

Then ∣∣∑

s

ν(L∗n,s)

∣∣ < c

n.

Since F is Λα-Carleson, we have∑

s

|L∗s|Λα(|L

∗s|) <∞,

and hence, ∑

s

ν(L∗n,s) →

∑

s

ν(L∗s)

as n→ ∞. Thus, ∑

s

ν(L∗s) = 0,

and hence, ν(F ) = 0. Again by Theorem 2.20, fν is cyclic in A−∞Λα

for α > α0. .

55

Chapitre 3

Cyclicity in weighted Bergman type

spaces

In this Chapter we study the cyclicity of S(z) = e−1+z1−z in the spaces B∞,0

Λ,E and BpΛ,E.Given a positive non-increasing continuous function Λ on (0, 1] and E ⊂ T = ∂D, we

denote by B∞Λ,E the space of all analytic functions f on D such that


|f(z)|e−Λ(dist(z,E)) < +∞,

and by B∞,0Λ,E its separable subspace

B∞,0Λ,E =

f ∈ B∞



Analogously, integrating with respect to area measure on the disc, we define the spacesBpΛ,E, 1 ≤ p <∞ :


∫

D


If E = T, we use the notation B∞Λ , B∞,0

Λ , BpΛ. Let us remark that either Λ(0+) = +∞or B∞

Λ = H∞, B∞,0Λ = 0, BpΛ = Bp0.

In the first part of this chapter we prove the following theorems :

Theorem 3.1 Let Λ be a positive non-increasing continuous function on (0, 1]. Then

S(z) = e−1+z1−z is cyclic in B∞,0

Λ if and only if Λ satisfies

∫

0

√Λ(t)

tdt = ∞. (3.1)

56

Theorem 3.2 Let 1 ≤ p < ∞ and let Λ be a positive non-increasing continuous function

on (0, 1]. Then S(z) = e−1+z1−z is cyclic in BpΛ if and only if Λ satisfies

∫

0

√Λ(t)

tdt = ∞.

3.1 Generalized Phragmén–Lindelöf Principle

We start with the Ahlfors–Carleman estimate of the harmonic measure (see, for example,[30, IX.E1]). Let G be a simply connected domain such that ∞ ∈ ∂G. Fix z0 ∈ G. Gi-ven ρ > 0 we denote by Gρ the connected component of the intersection of the discρD = z : |z| < ρ and G containing z0, Sρ is an arc on ∂(ρD) ∩G separating z0 from oneof the unbounded components of G \ ρD, s(ρ) is the length of Sρ. Then

ω(z0, Sρ, Gρ) ≤ C exp(−π

∫ ρ

0

dr

s(r)

)

for an absolute constant ; here ω(z0, ·,Ω) is the harmonic measure with respect to z ∈ Ωon the boundary of Ω.

Corollary 3.3 Suppose that G is as above, a function f is analytic in G, continuous upto ∂G \ ∞ and satisfies the conditions

|f(z)| ≤ 1, z ∈ ∂G,

lim infρ→∞

logM(ρ)

σ(ρ)= 0,

where

M(ρ) = maxz∈Sρ

|f(z)|, σ(ρ) = expπ

∫ ρ

1

dr

s(r)

.

Then |f(z)| ≤ 1, z ∈ G.

Given an increasing differentiable function φ : [0,+∞[→ R+ such that

limx→∞

φ(x)

x= ∞, (3.2)

we consider the domain Gφ defined by :

Gφ :=x+ iy ∈ C : |y| ≤ φ(x), x ≥ 0

.

57

Proposition 3.4 Let f be a function analytic on Gφ and continuous up to ∂Gφ \ ∞such that for some c > 0 we have

|f(z)| ≤ ec|z|, z ∈ Gφ,

|f(ξ)| ≤ 1, ξ ∈ ∂Gφ \ ∞.

If ∫ ∞ xφ′(x) dx

φ(x)2= +∞, (3.3)

then |f(z)| ≤ 1, z ∈ Gφ.

Proof. In the notations of Corollary 3.3, we have logM(ρ) ≤ cρ.A simple geometric argument shows that if

r2 = x2 + φ(x)2, (3.4)

thens(r) = r

(π − 2 arctan

x

φ(x)

).

Therefore, if x = x(r) is defined by (3.4), then

π

s(r)−

1

r≥

x

2rφ(x)≥

x

3φ(x)2

for large r (we use (3.2)). Moreover,

dr =x+ φ(x)φ′(x)

rdx ≥

φ′(x) dx

2

for large r.Thus, for some c > 0 we have

σ(ρ) ≥ cρ · exp[16

∫ x(ρ)

x(1)

xφ′(x) dx

φ(x)2

].

Therefore, (3.3) implies that

limρ→∞

logM(ρ)

σ(ρ)= 0,

and it remains to apply Corollary 3.3.

By the standard maximum principle, it suffices to require in Proposition 3.4 that φ(x)is increasing for large x.

Given a positive decreasing differentiable function Λ on (0, 1] such that Λ(0+) = +∞and

tΛ(t) = o(1), t→ 0, (3.5)

58

we consider the domain

ΩΛ =w ∈ D :

1− |w|2

|1− w|2≥ Λ(1− |w|2)

.

Let F (w) = 1+w1−w

be a conformal map of the unit disc onto the right half plane, and letx+ iy be a point on the boundary of F (ΩΛ), y ≥ 0. Then

x = Λ( 4x

(x+ 1)2 + y2

).

By monotonicity of Λ, y is determined uniquely by x, y = y(x), and we have F (ΩΛ) = Gy.Furthermore, we obtain that

4x2

(x+ 1)2 + y2=

4x

(x+ 1)2 + y2· Λ

( 4x

(x+ 1)2 + y2

)= o(1), x→ ∞.

Hence x = o(y), x→ ∞.Next, for sufficiently small positive t we have

4Λ(t)

(Λ(t) + 1)2 + y(Λ(t))2= t,

and, hence, y is differentiable and

2tΛ′(t)− Λ(t)

t2= Λ′(t)

(Λ(t) + 1 + y(Λ(t))y′(Λ(t))

).

Since tΛ(t) = o(1) and y(Λ(t)) = (2 + o(1))√Λ(t)/t, t→ 0, we obtain

Λ′(t)y′(Λ(t)) = (1 + o(1))tΛ′(t)− Λ(t)

t√tΛ(t)

, t→ 0. (3.6)

In particular, the function y(x) increases for large x.Finally, we estimate the integral

I =

∫ ∞ xy′(x)

y(x)2dx.

By (3.6) we have

I ≥ c+

∫

0

t

5·|tΛ′(t)− Λ(t)|

t√tΛ(t)

dt = c+1

5

∫

0

√Λ(t)

tdt+

1

5

∫

0

|Λ′(t)|

√t

Λ(t)dt.

Integrating by parts and using (3.5), one can easily verify that the integrals∫0

√Λ(t)/t dt

and∫0|Λ′(t)|

√t/Λ(t) dt converge simultaneously. Thus, I = ∞ if and only if (3.1) holds.

59

Corollary 3.5 Let f be analytic on the domain ΩΛ and continuous up to ∂ΩΛ \1, whereΛ is a positive decreasing differentiable function on (0, 1] satisfying Λ(0+) = +∞ and (3.5).If for some c > 0 we have

(a) |f(w)| ≤ ec1−|w|2

|1−w|2 , w ∈ ΩΛ,

(b) |f(ξ)| ≤ 1, ξ ∈ ∂ΩΛ \ 1,

and if ∫

0

√Λ(t)

tdt = ∞,

then |f(z)| ≤ 1, z ∈ ΩΛ.

3.2 Auxiliary estimates

Given λ ∈ D, we define the Privalov shadow Iλ,

Iλ =eiθ : | arg(λe−iθ)| <

1− |λ|

2

(see Figure 3.1), which is an arc of the circle T centered at λ|λ|

(the radial projection ofλ onto the unit circle T) of length |Iλ| = 1 − |λ|. Furthermore, we consider an auxiliaryfunction

fλ(z) = expcλ1− |λ|2

|1− λ|2

∫

Iλ

eiθ + z

eiθ − zdθ,

where

c−1λ =

∫

Iλ

1− |λ|2

|eiθ − λ|2dθ.

Figure 3.1 – Privalov shadow.

Then|fλ(λ)S(λ)| = 1,

60

where S(z) = exp[−(1 + z)/(1− z)]. Next, a geometric argument shows that

c−1λ ≥

1− |λ|2

5/4(1− |λ|)2· (1− |λ|) ≥

4

5,

and we have

supD

|fλ| ≤ exp(5π2

·1− |λ|2

|1− λ|2

).

Given a > 0, A > 1, we consider the domain ΓΛ,T(a,A) defined by

ΓΛ,T(a,A) =λ ∈ D :

1− |λ|2

|1− λ|2≤ aΛ(A(1− |λ|))

.

Now for some a,A we establish the following estimate :

Lemma 3.6sup

λ∈ΓΛ,T(a,A)

‖fλS‖Λ <∞. (3.7)

Proof. We set

Hλ(z) = cλ1− |λ|2

|1− λ|2

∫

Iλ

1− |z|2

|eiθ − z|2dθ −

1− |z|2

|1− z|2− Λ(1− |z|),

and obtain|fλ(z)S(z)|e

−Λ(1−|z|) = eHλ(z).

Thus, it remains to verify that

supλ∈ΓΛ,T(a,A), z∈D

Hλ(z) <∞.

Case 1 : If A(1− |λ|) ≥ 1− |z|, then

Hλ(z) ≤ 2πcλ1− |λ|2

|1− λ|2− Λ(1− |z|)

≤5πa

2Λ(A(1− |λ|))− Λ(A(1− |λ|)) =

(5πa2

− 1)Λ(A(1− |λ|)).

Therefore, for a ≤ 25π

we obtain that Hλ(z) ≤ 0.

Case 2 : If |1− z| ≥ 6|1− λ|, then for every eiθ ∈ I(λ) we have

|z − eiθ| ≥ |1− z| − |1− λ| − |λ− eiθ| ≥ |1− z| − |1− λ| − 2(1− |λ|)

≥ |1− z| − 3|1− λ| ≥1

2|1− z|.

61

Therefore,

Hλ(z) ≤ 51− |λ|2

|1− λ|21− |z|2

|1− z|2(1− |λ|)−

1− |z|2

|1− z|2≤

[5a(1− |λ|)Λ(A(1− |λ|))− 1

]1− |z|2

|1− z|2.

Since tΛ(t) = o(1), t→ 0, we obtain that Hλ(z) is uniformly bounded.

Case 3 : If A(1− |λ|) < 1− |z| and |1− z| < 6|1− λ|, then

1− |z|2

|1− z|2≥

A

72·1− |λ|2

|1− λ|2,

and hence,

Hλ(z) ≤180π

A·1− |z|2

|1− z|2−

1− |z|2

|1− z|2≤ 0,

for A ≥ 1000.

Now, let E be an arbitrary compact subset of T. For some a > 0, A > 1, we considerthe domain ΓΛ,E(a,A) defined by

ΓΛ,E(a,A) =λ ∈ D :

1− |λ|2

|1− λ|2≤ aΛ(A dist(λ,E))

,

and obtain the estimate

Lemma 3.7sup

λ∈ΓΛ,E(a,A)

‖fλS‖Λ <∞.

Proof. We just need to verify that

supλ∈ΓΛ,E(a,A), z∈D

Hλ(z) <∞.

In the cases (1) A dist(λ,E) ≥ dist(z, E) and (2) |1 − z| ≥ 6|1 − λ| we use the sameargument as in the proof of Lemma 3.6. In the case (3) we have A dist(λ,E) < dist(z, E)and |1−z| < 6|1−λ|. Take eiη ∈ E such that A|λ−eiη| ≤ |z−eiη| and use that for eiθ ∈ Iλwe have

1− |λ| ≤ |λ− eiθ| ≤ 2|λ− eiη|.

Then,

|z − eiθ| ≥ |z − eiη| − |λ− eiθ| − |λ− eiη| ≥ (A− 3)|λ− eiη| ≥A− 3

2(1− |λ|), eiθ ∈ E.

Therefore,

Hλ(z) ≤5

4

1− |λ|2

|1− λ|24

(A− 3)21− |z|2

|1− z|2(1− |λ|)−

1− |z|2

|1− z|2≤

( 360

(A− 3)2− 1

)1− |z|2

|1− z|2≤ 0

for A ≥ 100.

62

3.3 Proofs of Theorems 3.1 and 3.2

Proof of Theorem 3.1. Suppose that Λ is non-increasing and satisfies (3.1). ThenΛ(0+) = ∞, and slightly decreasing, if necessary, Λ, we can assume that Λ is decreasing.

If∫0Λ(t) dt = ∞, then the result follows from [17, Theorem 2]. Therefore, from now on

we assume that∫0Λ(t) dt <∞. Since Λ decreases, we have

Λ(t) ≤1

t

∫ t

0

Λ(s) ds = o(1t

), t→ 0,

and, hence, tΛ(t) = o(1), t→ 0. Finally, smoothing, if necessary, Λ, we can assume that Λis differentiable.

We denote by π the canonical projection of B∞,0Λ onto B∞,0

Λ /[S], and by α : z 7→ z theidentity map.

Suppose that 1 6∈ [S]. Next, we estimate ‖(λ − π(α))−1π(1)‖. Given λ ∈ D and ananalytic function f , we define the following function :

Lλ(f)(z) =

f(z)− f(λ)

z − λif z 6= λ

f ′(z) if z = λ .

For bounded functions f we have

f(λ)S(λ)π(1) = (λ− π(α))π(Lλ(fS)). (3.8)

In particular,S(λ)π(1) = (λ− π(α))π(Lλ(S)), λ ∈ C \ 1,

and hence, the function λ 7→ (λ− π(α))−1π(1) is well defined and analytic in C \ 1.Next we use that the function Q(z) = log |fλ(z)S(z)| is the Poisson integral

Q(z) = P(z, θ) ∗ dµ(θ)

of a finite (signed) measure

dµ(θ) = 2πcλ1− |λ|2

|1− λ|2χIλ(θ) dθ − δ1

with mass

|µ|(T) = 2πcλ(1− |λ|2)(1− |λ|)

|1− λ|2+ 1

uniformly bounded in λ. Since

supz∈D, θ∈T

(1− |z|)2‖∇P (z, θ)‖ <∞,

63

for an absolute constant c we obtain that

‖∇Q(z)‖ ≤c

(1− |z|)2.

Since Q(λ) = 0, we have |fλ(z)S(z)| ≤ c1 at the boundary of the disc Dλ = z : |z − λ| <(1− |λ|)2/2, and hence, by the maximum principle,

∣∣∣fλ(z)S(z)− fλ(λ)S(λ)

z − λ

∣∣∣ ≤ 2(c1 + 1)

(1− |λ|)2, z ∈ Dλ.

Therefore,

‖Lλ(fλS)‖Λ = sup|z|<1


z − λ

∣∣∣e−Λ(1−|z|)

≤ sup|z−λ|≤(1−|λ|)2/2


z − λ

∣∣∣e−Λ(1−|z|)

+ sup|z−λ|>(1−|λ|)2/2


z − λ

∣∣∣e−Λ(1−|z|)

≤2

(1− |λ|)2(‖fλS‖Λ + c1 + 2

).

By (3.8), for every λ ∈ D we have

‖(λ− π(α))−1π(1)‖Λ ≤1

|fλ(λ)S(λ)|‖Lλ(fλS)‖Λ ≤

2

(1− |λ|)2(‖fλS‖Λ + c1 + 2

). (3.9)

In the same way, by (3.8), for every λ ∈ D we have

‖(λ− π(α))−1π(1)‖Λ ≤1

|S(λ)|‖π(Lλ(S))‖Λ ≤

2

1− |λ|exp

1− |λ|2

|1− λ|2. (3.10)

Furthermore, for |λ| > 1 we have

‖(λ− π(α))−1π(1)‖Λ ≤1

|S(λ)|‖π(Lλ(S))‖Λ ≤

2

|λ| − 1. (3.11)

Let us fix a and A such that (3.7) holds. For λ ∈ ∂ΓΛ(a,A) \ T we have 1 − |λ|2 =aΛ(A(1− |λ|))|1− λ|2, and hence,

1

1− |λ|≤

c

|1− λ|2, λ ∈ ∂ΓΛ(a,A) \ T. (3.12)

By Lemma 3.6 and (3.9), we have

supλ∈∂ΓΛ(a,A)

‖(λ− 1)4(λ− π(α))−1π(1)‖ <∞. (3.13)

64

Applying (3.10) and Corollary 3.5 (to Λ1 defined by Λ1(1− r2) = aΛ(A(1− r))), we obtainthat the function (λ− 1)4(λ− π(α))−1π(1) is bounded on D \ ΓΛ(a,A) = IntΩΛ1 .

By (3.12), for some c > 0 we have

dist(ζ, ∂ΓΛ(a,A) \ T) ≥ c|1− ζ|2, ζ ∈ T.

Applying Levinson’s log-log theorem (see, for example, [30, VII D7]) or, rather, its polyno-mial growth version to the function (λ−π(α))−1π(1) and using estimates (3.13) and (3.11)we obtain that

‖(λ− π(α))−1π(1)‖ ≤C

|λ− 1|4, λ ∈ 2D \ (1 ∪ D \ ΓΛ(a,A)). (3.14)

By (3.13) and (3.14), the function λ 7→ (λ− 1)4(λ− π(α))−1π(1) is bounded on 2D \ 1.Now, pick an arbitrary functional φ⊥[S], and consider the function

Φ(λ) = (λ− 1)〈(λ− π(α))−1π(1), φ〉.

The function Φ is analytic on C \ 1 and has a pole of order at most 3 at the point 1. By(3.11), we have

lim supε→0+

|Φ(1 + ε)| <∞,

and hence, Φ is an entire function. Again by (3.11), Φ is bounded and hence is a constantfunction. Furthermore,

Φ(λ) = 〈(λ−π(α))−1π(λ−1), φ〉 = 〈(λ−π(α))−1π(λ−α), φ〉+ 〈(λ−π(α))−1π(α−1), φ〉

= 〈π(1), φ〉+ 〈(λ− π(α))−1π(α− 1), φ〉.

Arguing as in the proof of (3.11), we see that

lim|λ|→∞

(λ− π(α))−1π(α− 1) = 0,

and hence, 〈α − 1, φ〉 = 0. Since φ is an arbitrary functional vanishing on [S], we getα − 1 ∈ [S], αn − 1 ∈ [S], n ≥ 1, and hence, 1 ∈ [S]. This contradiction shows that[S] = B∞,0

Λ .In the opposite direction, suppose that S is cyclic in B∞,0

Λ . We replace Λ by Λ∗,

Λ∗(t) = Λ(t) + log1

t,

and remark that S is also cyclic in B∞,0Λ∗ . Now we can apply the result by Nikolski [39,

Theorem 1a, Section 2.6] to conclude that Λ∗ satisfies (3.1).

Proof of Theorem 3.2. It suffices to remark that

B∞,0Λ ⊂ BpΛ ⊂ B∞,0

Λ∗ ,

65

where

Λ∗(t) = Λ(t/2) +2

plog

1

t,

and to apply Theorem 3.1. We use here the subharmonicity of |f |p and the fact that Λ andΛ∗ satisfy (3.1) simultaneously.

Remark 3.8 The same method works for the spaces BpΛ,E, where E is a closed arc of theunit circle. More precisely we have the following result :

Let E be a non-trivial closed arc of the unit circle such 1 ∈ E, let 1 ≤ p < ∞, and let

Λ be a positive non-increasing continuous function on (0, 1]. Then S(z) = e−1+z1−z is cyclic

in BpΛ,E if and only if Λ satisfies (3.1).

Sketch of the proof. For the necessity part it suffices to use that BpΛ,E ⊂ BpΛ. In the

opposite direction, suppose that Λ satisfies (3.1) and let E = (eia, eib). If 1 ∈ (eia, eib) thenthe same proof works. In the case eia = 1 a slight modification is needed : we replace ΩΛ

by

ΩΛ =w ∈ D : Imw ≥ 0,

1− |w|2

|1− w|2≥ aΛ(A(1− |w|2))

∪w ∈ D : Imw ≤ 0,

1− |w|2

|1− w|2≥ a1Λ(A1|1− w|)

for some a, a1, A,A1.

3.4 An auxiliary domain for general E

Let E be a compact subset of T, 1 ∈ E. Given a positive decreasing C1 smooth functionΛ on (0, 1) such that Λ(1) < 1/10, and

limt→0

Λ(t) = ∞, limt→0

tΛ(t) = 0, t|Λ′(t)| = O(Λ(t)), t→ 0, (3.15)

we consider the domain

ΩΛ,E = (1− s)eiθ : s ≥ θ2Λ(s+ dist(eiθ, E)), θ ∈ (−π, π].

Clearly, ΩΛ,E is star-shaped with respect to the origin,

∂ΩΛ,E = (1− γ(θ))eiθ, θ ∈ (−π, π]

withγ(θ) = θ2Λ(γ(θ) + dist(eiθ, E)). (3.16)

66

Next,

infθ∈(−π,π]\0

γ(θ)

θ2> 0, (3.17)

limθ→0 γ(θ) = 0, and

γ(θ)

θ= [γ(θ)Λ(γ(θ) + dist(eiθ, E))]1/2 = o(1), θ → 0.

Furthermore, the derivative h(θ) of dist(eiθ, E) is equal to ±1 for a.e. eiθ on T \ E and to0 for a.e. eiθ on E. Therefore,

γ′(θ) = 2θΛ(γ(θ) + dist(eiθ, E))− θ2|Λ′(γ(θ) + dist(eiθ, E))|(γ′(θ) + h(θ))

for a.e. eiθ ∈ T, and hence, by (3.15),

|γ′(θ)| = O(1), a.e. θ → 0. (3.18)

Now we set

(1− γ(θ))eiθ = 1−eiφ

R,

with φ ∈ [−π/2, π/2]. Then

R ≍1

θ,

π

2− |φ| ≍

γ(θ)

θ,

dR

dθ= (1 + o(1))R2, θ → 0.

By analogy with Proposition 3.4 and Corollary 3.5 we have

Proposition 3.9 Given a continuous function φ : R+ → (0, π/2) let

Gφ :=Reiθ : |θ| <

π

2− φ(R), R > 0

.

Let f be a function analytic on Gφ and continuous up to ∂Gφ \ ∞ such that for somec > 0 we have

|f(z)| ≤ ec|z|, z ∈ Gφ,

|f(ξ)| ≤ 1, ξ ∈ ∂Gφ \ ∞.

If ∫ ∞ φ(R) dR

R= +∞, (3.19)

then |f(z)| ≤ 1, z ∈ Gφ.

67

Corollary 3.10 Let f be analytic on the domain ΩΛ,E and continuous up to ∂ΩΛ,E \ 1,where Λ is a positive decreasing differentiable function on (0, 1] satisfying (3.15). If forsome c > 0 we have

(a) |f(w)| ≤ ec1−|w|2

|1−w|2 , w ∈ ΩΛ,E,

(b) |f(ξ)| ≤ 1, ξ ∈ ∂ΩΛ,E \ 1,

and if ∫

0

γ(θ)

θ2dθ = +∞,

then |f(z)| ≤ 1, z ∈ ΩΛ,E.

Later on, we need the following result.

Proposition 3.11 Let Λ be a positive decreasing differentiable function on (0, 1] satisfying(3.15), and let ∫

0

γ(θ)

θ2dθ < +∞. (3.20)

There exists an outer function F such that

|F (w)| > e1−|w|2

|1−w|2+Λ(dist(w,E))

, w ∈ ∂ΩΛ,E \ 1. (3.21)

Proof. By (3.20), we can set

log |F (eiθ)| = Aγ(θ)

θ2

for some A to be chosen later.Given w = (1− γ(θ))eiθ ∈ ∂ΩΛ,E and β > 0 we have

log |F (w)| ≥ β1 min|ψ−θ|<βγ(θ)

γ(ψ)

ψ2

with β1 = β1(β) > 0. Furthermore, by (3.18), for some β > 0 we have

min|ψ−θ|<βγ(θ)

γ(ψ)

ψ2≥γ(θ)

2θ2.

Now, (3.21) follows for sufficiently large A.

68

3.5 General E

Let Λ be a positive decreasing differentiable function on (0, 1] satisfying (3.15), and letE be a compact subset of T, 1 ∈ E. We define γ by (3.16).

Theorem 3.12 The function S(z) = e−1+z1−z is cyclic in B∞,0

Λ,E if and only if the integral

∫γ(θ)

θ2dθ (3.22)

diverges at 0.

Proof. If the integral diverges, we use the same method as in the proof of Theorem 3.1 ;we use Corollary 3.10 instead of Corollary 3.5 and Lemma 3.7 instead of Lemma 3.6. Theestimate (3.12) is replaced by (3.17).

In the opposite direction, if the integral converges, we use Proposition 3.11 and provethat S is not cyclic by the Keldysh method (see [25], [39, Section 2.8.2]).

From now on we assume that

Λ(t) =1

tw(t)2, (3.23)

where w is a positive decreasing C1 smooth function on (0, 1), limt→0w(t) = +∞, w(t2) ≍w(t), |w′(t)| = O(w(t)/t), t→ 0. Such Λ satisfy (3.15). Typical w are logp(1/t), p > 0.

Remark 3.13 For such Λ, the theorem of Nikolski [39] implies that if∫0

dt|t|w(|t|)

< ∞,

then S is not cyclic in B∞,0Λ,E ; the theorem of Gevorkyan–Shamoyan [20] implies that if∫

0dt

|t|w(|t|)2= ∞, then S is cyclic in B∞,0

Λ,E .

Let In be the arcs complementary to E, In = (eian , eibn) or In = (e−ibn , e−ian), 0 < an <bn. We divide the family of all such arcs into three groups :

the short intervals : I1 =In : 1− an

bn< 2

w(bn)

,

the intermediate intervals : I2 =In : 2

w(bn)≤ 1− an

bn< 1

2

,

and the long intervals : I3 =In : an

bn≤ 1

2

.

Theorem 3.14 Let Λ be defined by (3.23) with w satisfying the above conditions. The


Λ,E if and only if

∫

eit∈E ∪⋃

In∈I1In

dt

|t|w(|t|)+

∑

In∈I2

1

w(bn)2log

[(1−

anbn

)w(bn)

]

+∑

In∈I3

logw(bn)

w(bn)2+

∫

eit∈⋃

In∈I3In

dt

|t|w2(|t|)= +∞. (3.24)

69

Remark 3.15 One can easily verify that condition (3.24) is equivalent to the divergenceof at least one of the following three expressions :

∫

eit∈E

dt

|t|w(|t|),

∫

0

dt

|t|w2(|t|),

∑

n

1

w(bn)2log

[1 +

(1−

anbn

)w(bn)

],

where the sums runs by all the arcs In complementary to E.

Proof. By Theorem 3.12, we just need to study the convergence of the integral∫γ(t)

t2dt

at 0.(a). For eit ∈ E, t > 0, we have

γ(t)

t2= Λ(γ(t)) =

1

γ(t)w(γ(t))2,

and hence,

γ(t)w(γ(t)) = t,

γ(t) ≍t

w(t),

γ(t)

t2≍

1

tw(t). (3.25)

Here we use that under our conditions on w, the function inverse to t 7→ tw(t) is equivalentto t 7→ t/w(t).

(b). Let eit ∈ I = eis : 0 < a < s < b ∈ I1. (The case b < t < a < 0 is treatedanalogously.) Then

γ(t)

t2= Λ(γ(t) + dist(eit, E)) ≤ Λ(γ(t))

and

dist(eit, E) < |b− a| <2b

w(b).

Hence,

γ(t) .t

w(γ(t)).

t

w(t),

and

Λ(γ(t) + dist(eit, E)) & Λ(γ(t)),

γ(t) &t

w(γ(t))&

t

w(t).

70

Finally,γ(t)

t2≍

1

tw(t). (3.26)

(c). Let eit ∈ I = eis : 0 < a < s < b ∈ I2. We have

γ(t)

t2≍

1

(γ(t) + dist(t, a, b))w(b)2,

and hence,

γ(t)(γ(t) + dist(t, a, b)) ≍b2

w(b)2.

Therefore, for some c > 0 and for

b−b

cw(b)< t < b

we have

γ(t) ≍b

w(b),

∫ b

b− bcw(b)

γ(t)

t2dt ≍

1

w(b)2,

and fora+ b

2< t < b−

b

cw(b)

we have

γ(t) ≍b2

w(b)2 dist(t, a, b),

∫ b− bcw(b)

(a+b)/2

γ(t)

t2dt ≍

1

w(b)2log

[c(1−

a

b

)w(b)

].

The integral from a to (a+ b)/2 is estimated in an analogous way, and we get∫ b

a

γ(t)

t2≍

1

w(b)2log

[(1−

a

b

)w(b)

]. (3.27)

(d). Let I = eis : 0 < a < s < b ∈ I3. As in part (c), the integral∫ b

(a+b)/2

γ(t)

t2

is equivalent tologw(b)

w(b)2.

71

Next, ∫ a+ aw(a)

a

γ(t)

t2≍

1

w(a)2.

For t ∈ (a+ aw(a)

, a+b2) we have

γ(t)

t2= Λ(γ(t) + (t− a)) ≤ Λ(γ(t)),

andγ(t) ≤

t

w(γ(t)).

t

w(t).

Therefore,γ(t)

t2≍ Λ(t− a),

and ∫ (a+b)/2

a+ aw(a)

γ(t)

t2dt ≍

∫ (b−a)/2

aw(a)

dt

tw(t)2≍

∫ b

a

dt

tw(t)2+

logw(a)

w(a)2.

Thus, ∫ b

a

γ(t)

t2dt ≍

∫ b

a

dt

tw(t)2+

logw(b)

w(b)2. (3.28)

The theorem follows from (3.25)–(3.28).

Corollary 3.16 Let Λ be as in the formulation of Theorem 3.14. This theorem yieldsimmediately that if E = exp(i · 2−n)n≥1 ∪ 1, then S is cyclic in B∞,0

Λ,E if and only if

∑

n≥1

logw(2−n)

w(2−n)2= +∞ ;

if E = exp(i · 2−2n)n≥1 ∪ 1, then S is cyclic in B∞,0Λ,E if and only if

∫

0

dt

tw(t)2= +∞,

and we return to the Gevorkyan–Shamoyan condition valid for E = 1.

Next we give two more applications of the general criterion (3.24).Let us introduce a condition

∫

0

Λ(t)1−β

tβdt = +∞, 0 ≤ β ≤

1

2(Cβ)

interpolating between that by Nikolski (∫0

√Λ(t)/t dt = +∞, β = 1/2) and that by

Gevorkyan–Shamoyan (∫0Λ(t) dt = +∞, β = 0).

72

For

Λα(t) =1

t logα(1/t)

we haveΛα ∈ (Cβ) ⇐⇒ α(1− β) ≤ 1.

Theorem 3.17 Let 0 ≤ β ≤ 1/2, an = exp(−n1−β), n ≥ 1, Eβ = eiann≥1 ∪ 1. The


Λα,Eβif and only if Λα ∈ (Cβ).

Proof. We have

1−an+1

an= 1− exp

[n1−β − (n+ 1)1−β

]≍ n−β.

Consider three cases.(a). α(1− β) > 1. Then all the arcs eitan+1<t<an are intermediate ones, and we need

only to verify that

∑

n≥1

1

logα(exp(n1−β))log

[n−β logα/2(exp(n1−β))

]≍

∑

n≥1

log(n−β+α(1−β)/2)

nα(1−β)< +∞.

(b). 2β < α(1− β) ≤ 1. Again all the arcs eitan+1<t<an are intermediate ones, and

∑

n≥1

1

logα(exp(n1−β))log

[n−β logα/2(exp(n1−β))

]≍

∑

n≥1

log(n−β+α(1−β)/2)

nα(1−β)= +∞.

(c). α(1−β) ≤ 2β ≤ 1. In this case we can assume that all the arcs are short ones, andwe have ∫

0

dt

t logα/2(1/t)= +∞.

Together, (a), (b), and (c) prove the assertion of the theorem.

Finally, we deal with the Cantor ternary set F . Let F0 = [0, 1]. On step n ≥ 0, Fnconsists of 2n intervals Ij = [aj, bj]. We divide each of them into three equal subintervals

Ij =[aj,

2aj + bj3

]∪[2aj + bj

3,aj + 2bj

3

]∪[aj + 2bj

3, bj

]

and setFn+1 =

⋃

j

I1j ∪⋃

j

I3j .

We define F = ∩n≥1Fn. Denote by κ the Hausdorff dimension of F (see [18, Section 1.5]),κ = log 2

log 3.

73

Theorem 3.18 Let E = eit : t ∈ F. The function S(z) = e−1+z1−z is cyclic in B∞,0

Λα,Eif

and only if

α ≤1

1− κ2

.

Proof. The set E is of zero measure ; all the complementary arcs are short or interme-diate. For simplicity, we pass to F ⊂ [0, 1]. For every N ≥ 1 we have 2N complementaryintervals of length 3−N . In every interval [3s−N , 2 · 3s−N ] we have 2s of such intervals,0 ≤ s < N . They are short for 3−s(N − s)α/2 . 1 and intermediate for 3−s(N − s)α/2 & 1.

The sum for the intermediate intervals in (3.24) is

∑

N≥1

∑

s≥0, 3−s(N−s)α/2&1

∑

In=[an,bn]⊂[3s−N ,2·3s−N ]

logα1

bnlog

[bn − anbn

logα/21

bn

]

≍∑

N≥1

∑

s≥0, 3−s(N−s)α/2&1

log+(3−s(N − s)α/2)

(N − s)α· 2s ≍

∑

N≥1

Nακ/2

Nα;

the latter series diverges if and only if α(1− κ2) ≤ 1.

The integral for the short intervals in (3.24) is

∑

N≥1

∑

s≥0, 3−s(N−s)α/2.1

∑

In=[an,bn]⊂[3s−N ,2·3s−N ]

∫ bn

an

dt

tw(t)

≍∑

N≥1

∑

s≥0, 3−s(N−s)α/2.1

∑

In=[an,bn]⊂[3s−N ,2·3s−N ]

1

(N − s)α/2log

bnan

≍∑

N≥1

∑

s≥0, 3−s(N−s)α/2.1

1

Nα/2·2s

3s≍

∑

N≥1

N (α/2)(κ−1)

Nα/2;

the latter series diverges if and only if α(1− κ2) ≤ 1.

Since the Cantor set F is self-similar, we get the same result for every shift of E :Ex = ei(y−x) : eiy ∈ E, eix ∈ E. On the other hand, it looks difficult to characterize thethreshold value of α in terms of (the local behavior near the point 1) for general sets E.

74

Chapitre 4

References

75

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